On the Study of Nonlinear Capillarity Models Through Weighted Sobolev Spaces with Dirichlet Boundary Conditions

Abstract

We prove in this paper some existence and uniqueness results of weak solutions for some nonlinear degenerate parabolic problems arising from capillarity phenomena with the Dirichlet type boundary condition.

1. Introduction

In this paper, we establish existence and uniqueness results for weak solutions to a class of nonlinear degenerate parabolic problems arising from capillarity phenomena, subject to Dirichlet-type boundary conditions:

$$\left\{\begin{array}{cccc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}-div\omega \left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u+\omega {\left|u\right|}^{p-2}u=f,\hfill & \hfill & \mathrm{in}Q=\Omega \times (0,T],\hfill \\ \hfill & u=0,\hfill & \hfill & \mathrm{on}\Gamma =\partial \Omega \times (0,T),\hfill \\ \hfill & {u|}_{t=0}=u_0,\hfill & \hfill & \mathrm{in}\Omega ,\hfill \end{array}\right.$$

(1)

where $\Omega$ is a bounded open subset of ${\mathbb{R}}^N$ with $N\ge 2$, $T>0$, $p>1$; $\nabla u$ is the gradient of u; f is a function defined on Q; and $\omega$ is a weight function (i.e., a locally integrable function on ${\mathbb{R}}^N$ such that $0<\omega \left(x\right)<\infty$ almost everywhere in ${\mathbb{R}}^N$), satisfying certain regularity assumptions detailed in Section 2.

The capillarity problem in fluid mechanics concerns the configuration of a liquid interface adjacent to a solid boundary, governed by surface tension and external forces such as gravity [1,2]. A classical example is the rise of a liquid in a thin vertical tube—a phenomenon that also manifests in natural processes such as water transport in soil and capillary action in plants [1,3]. Capillarity arises from the interplay between two fundamental molecular forces: adhesion, which characterizes the interaction between the liquid and the solid surface, and cohesion, which governs intermolecular attraction within the liquid [4]. Understanding this balance is essential for describing the shape and stability of liquid interfaces [2,5]. In recent years, there has been renewed interest in capillary effects due to their importance in various physical and engineering contexts, including droplet and bubble dynamics, wetting and spreading phenomena, and applications in industrial, biomedical, and microfluidic systems [3,6,7,8].

Given the practical importance of capillary flows, it is crucial to construct mathematical models that accurately describe their behavior. Such problems are frequently formulated as boundary value problems for nonlinear partial differential equations. The classical model for capillary rise, known as the Washburn Equation [9], provides a simplified description of the dynamics of liquid ascent in narrow tubes. However, experimental evidence shows that for certain fluids, the free surface oscillates near the equilibrium level rather than stabilizing monotonically.

From a physical standpoint, the nonlinear diffusion operator $-div\left({\omega \left(x\right)|\nabla u|}^{p-2}\nabla u\right)$ is used to describe how things like pressure, heat, or chemicals spread in materials where the properties change from place to place. The function u shows the quantity that moves (like temperature or concentration), and $\omega \left(x\right)$ tells how easily it moves in different parts of the material. The number p shows if the spreading is normal ($p=2$) or nonlinear ($p\ne 2$).

The formula for the flow (or flux) is ${\omega \left(x\right)|\nabla u|}^{p-2}\nabla u$, which is a more general version of Darcy’s law. This operator is used in models of porous media [10], fluids that react to electric fields [11], and many other systems. Its origin from physical laws like fluid flow and conservation of mass is clearly explained in [12].

A representative example of a weight satisfying the required structural assumptions is $\omega \left(x\right)={\left|x\right|}^{\alpha }$ with $\alpha >-N$, which describes a medium whose permeability varies radially or degenerates near the origin. In the context of capillarity, such a weight models spatially nonuniform materials where surface tension effects depend on position—e.g., porous or rough surfaces whose microstructure alters the local capillary pressure. This interpretation provides a physical rationale for incorporating spatially dependent weights into the diffusion term of (1).

In parallel to the p-Laplacian-type operator, another important nonlinear diffusion operator arises in capillarity theory, particularly in the foundational works by Ni and Serrin [13,14]. They studied stationary solutions of the equation

$$-div\left({\displaystyle \frac{\nabla u}{1+{|\nabla u|}^2}}\right)=f\left(u\right),\mathrm{in}{\mathbb{R}}^N,$$

which emerges in models of prescribed mean curvature surfaces and capillary interfaces. Unlike the p-Laplacian, where the nonlinearity depends on the gradient magnitude via a power law, this operator captures the saturation effect of fluxes in high-gradient regimes and reflects the geometry of the interface itself. When $f\left(u\right)=0$, it reduces to the minimal surface equation.

This form naturally appears in the modeling of phenomena such as soap films, capillary rise in porous media, and thin liquid layers where curvature-driven effects dominate. Its geometric character, in contrast to the energy-based structure of the p-Laplacian, makes it central to the analysis of curvature-sensitive transport problems [2].

Since then, considerable research has been devoted to understanding capillarity-driven problems. For example, Rodrigues [15] employed variational methods, including the mountain pass lemma and the fountain theorem, to establish the existence of nontrivial solutions for the following problem:

$$\left\{\begin{array}{cccc}\hfill & -div\left({|\nabla u|}^{p\left(x\right)-2}\nabla u+{\displaystyle \frac{{|\nabla u|}^{2p\left(x\right)-2}\nabla u}{\sqrt{1+{|\nabla u|}^{2p\left(x\right)}}}}\right)=\lambda f(x,u),\hfill & \hfill & \mathrm{in}\Omega ,\hfill \\ \hfill & u=0,\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(2)

where $\Omega \subset {\mathbb{R}}^N$ ($N\ge 2$) is a bounded regular domain, $\lambda >0$ is a parameter, and f is a Carathéodory function.

Furthermore, Shokooh et al. [16] investigated the existence of three weak solutions for the following problem:

$$\left\{\begin{array}{cccc}\hfill & -div\left(1+{\displaystyle \frac{{|\nabla u|}^{2p\left(x\right)-2}}{\sqrt{1+{|\nabla u|}^{2p\left(x\right)}}}}\right)\nabla u+a\left(x\right){\left|u\right|}^{p\left(x\right)-2}u=\lambda f(x,u)+\mu g(x,u),\hfill & \hfill & \mathrm{in}\Omega ,\hfill \\ \hfill & {\displaystyle \frac{\partial u}{\partial \nu }}=0,\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(3)

where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with $C^1$ boundary; $\nu$ is the outward unit normal to $\partial \Omega$, $\lambda >0$, $\mu \ge 0$; and $a\in L^{\infty }(\Omega )$ with $a\ge 0$ a.e. The functions $f,g:\Omega \times \mathbb{R}\to \mathbb{R}$ are $L^1$-Carathéodory functions, and the exponent $p\in C\left(\overline{\Omega }\right)$ satisfies the condition

$$N<p_{-}:=\underset{x\in \overline{\Omega }}{inf}p\left(x\right)\le p_{+}:=\underset{x\in \overline{\Omega }}{sup}p\left(x\right)<\infty .$$

The present work offers new theoretical insights into capillarity-type diffusion models by extending known elliptic results to a fully time-dependent, degenerate parabolic framework with weighted nonlinearities. The operator in (1) captures both nonlinear diffusion and saturation phenomena through a combination of p-Laplacian and higher-order capillarity terms, modulated by spatially varying weights.

In contrast to previous studies by Rodrigues [15] and Shokooh et al. [16], which addressed only elliptic problems, our analysis establishes the existence and uniqueness of weak solutions to the corresponding parabolic problem. The proof is based on Rothe’s method, which applies a semi-discretization in time to reduce the problem to a sequence of nonlinear elliptic subproblems. At each time step, we consider the following discretized formulation:

$$\left\{\begin{array}{cc}\hfill & {\displaystyle \frac{u_k-u_{k-1}}{h}}-div\omega \left(|\nabla u_k{|}^{p-2}+{\displaystyle \frac{|\nabla u_k{|}^{2p-2}}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right)\nabla u_k+\omega {\left|u_k\right|}^{p-2}{u}_k={\left[f\right]}_h(x,(k-1)h),\mathrm{in}\Omega ,\hfill \\ \hfill & u_k=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(4)

where $h>0$, k is positive integer and

$${\left[f\right]}_h(x,t):={\displaystyle \frac{1}{h}}{\int }_t^{t+h}f(x,s)ds.$$

This discretized formulation defines a nonlinear elliptic problem for $u_k$, for which the monotone operator theory ensures the existence of a unique weak solution $u_k$. Based on these discrete solutions, we construct an approximate solution sequence and establish some a priori estimates; by extracting a subsequence, we obtain a limit function, which we then prove to be a weak solution of the original parabolic problem. This framework strengthens the theoretical understanding of degenerate parabolic systems and supports the modeling of nonlinear capillarity and diffusion in spatially heterogeneous media.

This paper is organized as follows. In Section 2, we present preliminary results and technical tools necessary for the analysis. In Section 3, we establish the existence of weak solutions to the semi-discrete problem (4). Finally, in Section 4, we extend the result to show the existence and uniqueness of weak solutions to the original problem (1) via the semi-discretization approach.

2. Preliminaries and Notations

This section provides notations, definitions, and resuls that will be utilized in this work. Let $\Omega$ denote a smooth bounded domain in ${\mathbb{R}}^N$. A weight refers to a locally integrable function $\omega$ on ${\mathbb{R}}^N$ satisfying $0<\omega <\infty$ for almost every $x\in {\mathbb{R}}^N$. We will designate by $L^p\left(\Omega ,\omega \right)$ the collection of all measurable functions u on $\Omega$ with a finite norm.

$${\parallel u\parallel }_{L^p\left(\Omega ,\omega \right)}={\left({\int }_{\Omega }\omega \left(x\right){\left|u\right|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}},1\le p<\infty .$$

The weighted Sobolev space $W^{1,p}\left(\Omega ,\omega \right)$ is defined as the set of all functions $u\in L^p\left(\Omega \right)$ whose weak derivatives $\nabla u\in L^p\left(\Omega ,\omega \right)$, endowed with the finite norm.

$$\begin{array}{cc}\hfill {\parallel u\parallel }_{W^{1,p}\left(\Omega ,\omega \right)}& :={\parallel u\parallel }_{L^p\left(\Omega \right)}+{\parallel \nabla u\parallel }_{L^p{\left(\Omega ,\omega \right)}^N}.\hfill \end{array}$$

The space of all functions with continuous derivatives of arbitrary order and compact support in $\Omega$ is denoted by $C_0^{\infty }(\Omega )$, and the space $W_0^{1,p}(\Omega ,\omega )$ denotes the closure of $C_0^{\infty }(\Omega )$ in $W^{1,p}(\Omega ,\omega )$, equipped with the norm

$${\parallel u\parallel }_{W_0^{1,p}(\Omega ,\omega )}:={\left({\int }_{\Omega }\omega \left(x\right){|\nabla u\left(x\right)|}^{p}dx\right)}^{1/p},$$

where $\omega$ is a positive weight function.

Let $1\le p<\infty$, $(a,b)\subset \mathbb{R}$, and X be a Banach space. The space $L^p(a,b;X)$ is the set of measurable functions

$$f:(a,b)\to X$$

such that

$${\parallel f\parallel }_{L^p(a,b;X)}:={\left({\int }_a^b{\parallel f\left(t\right)\parallel }_X^{p}dt\right)}^{1/p}<\infty .$$

In this study, we presuppose that $\omega$ meets the following criteria:

• $\left(H_1\right)$$\omega \in L_{loc}^1\left(\Omega \right)$, ${\omega }^{{\displaystyle \frac{-1}{p-1}}}\in L_{loc}^1\left(\Omega \right),$
• $\left(H_2\right)$${\omega }^{-s}\in L^1\left(\Omega \right)$ where $s\in \left({\displaystyle \frac{N}{p}},\infty \right)\cap \left[{\displaystyle \frac{1}{p-1}},\infty \right)$.

Proposition 1

([17]). Assume that the hypotheses ($H_1$) and ($H_2$) hold. Then, for $s+1⩽ps<N(s+1)$, we have the continuous embedding

$$W_0^{1,p}\left(\Omega ,\omega \right)\hookrightarrow W_0^{1,p_1}(\Omega )\hookrightarrow L^q(\Omega ),$$

(5)

where $p_1={\displaystyle \frac{ps}{1+s}}$, $1\le q={\displaystyle \frac{N{p}_1}{N-p_1}}={\displaystyle \frac{Nps}{N(s+1)-ps}},$ and for $ps⩾N(s+1)$ the embedding (5) holds with arbitrary $1\le q<\infty$. Moreover, the compact embedding

$$W_0^{1,p}(\Omega ,\omega )\hookrightarrow L^r(\Omega ).$$

holds provided $1\le r<q.$

Proposition 2

([17] (Hardy-type inequality)). There exist a weight function ω on Ω and a parameter $q,1<q<\infty$ such that the inequality

$${\left({\int }_{\Omega }\omega {\left|u\left(x\right)\right|}^{q}dx\right)}^{{\displaystyle \frac{1}{q}}}⩽C{\left({\int }_{\Omega }\omega {|\nabla u|}^{p}dx\right)}^{{\displaystyle \frac{1}{p}}},$$

(6)

holds for every $u\in W_0^{1,p}(\Omega ,\omega )$ with a constant $C>0$ independent of u and, moreover, the embedding

$$W_0^{1,p}(\Omega ,\omega )\hookrightarrow L^q(\Omega ,\omega )$$

determined by the inequality (6) is compact.

Theorem 1

([18]). Let Y be a reflexive real Banach space, $Y^{\prime }$ its dual space, and $A:Y\to Y^{\prime }$ be a bounded, hemicontinuous, coercive, and monotone operator. Then the equation $Au=v$ has at least one solution $u\in Y$ for each $v\in Y^{\prime }$.

Lemma 1

([19]). Let $1<p<\infty$. There exist two positive constants ${\alpha }_p$ and ${\beta }_p$ such that for every $(\xi ,\eta \in {\mathbb{R}}^N)$ with $N\ge 1,$

$${\alpha }_p\left(\right|\xi |+{\left|\eta \right|)}^{p-2}{|\xi -\eta |}^2\le {⟨\left|\xi \right|}^{p-2}\xi -{\left|\eta \right|}^{p-2}\eta ,\xi -\eta ⟩\le {\beta }_p\left(\right|\xi |+{\left|\eta \right|)}^{p-2}|\xi -\eta |.$$

Lemma 2

(Lemma 2.2, [20]). For $\xi ,\eta \in {\mathbb{R}}^{N}and1<p<\infty ,$ we have

$$\left({\displaystyle \frac{{\left|\xi \right|}^{p-2}}{\sqrt{1+{\left|\xi \right|}^p}}}\xi -{\displaystyle \frac{{\left|\eta \right|}^{p-2}}{\sqrt{1+{\left|\eta \right|}^p}}}\eta \right)·\left(\xi -\eta \right)\ge 0.$$

For additional information regarding weighted Sobolev spaces, we direct the reader to [21,22,23].

3. Discretization of the Problem

To discuss the solution of (4), we focus on the given elliptic equation

$$\left\{\begin{array}{ccc}\hfill & {\displaystyle \frac{u-u_0}{h}}-div\omega \left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u+\omega {\left|u\right|}^{p-2}u={\left[f\right]}_h(x,0),\hfill & \hfill \mathrm{in}\Omega ,\\ \hfill & {\left.u\right|}_{\partial \Omega }=0.\hfill \end{array}\right.$$

(7)

Definition 1.

A function u is called a weak solution of the Problem (7) if $u\in W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega )$ and

$$\begin{array}{c}{\int }_{\Omega }{\displaystyle \frac{u-u_0}{h}}\varphi dx+{\int }_{\Omega }\omega \left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u·\nabla \varphi dx+{\int }_{\Omega }\omega {\left|u\right|}^{p-2}u\varphi dx\hfill \\ ={\int }_{\Omega }{\left[f\right]}_h(x,0)\varphi dx,\hfill \end{array}$$

(8)

for all $\varphi \in W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega ).$

Theorem 2.

Let $\left({\left[f\right]}_h(·,0)/\omega \right)\in L^{p^{\prime }}(\Omega ,\omega )$, and assume that hypotheses $\left(H_1\right)$ and $\left(H_2\right)$ hold. Then problem (7) has a unique weak solution.

Proof. 

Existence of the weak solutions:

Let

$$E:=W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega ),$$

which is a Banach space equipped with the following norm

$${\left|\right|u\left|\right|}_E:={\left({\left|\right|u\left|\right|}_{W_0^{1,p}(\Omega ,\omega )}^p+{\left|\right|u\left|\right|}_{L^2(\Omega )}^2\right)}^{1/p}.$$

We consider the operator $A:E\to E^{\prime }$ such that

$$A=A^0+A^1+A^2+A^3-A^4,$$

where for all $u,vs.\in E$

$$\begin{array}{cc}\hfill <A^{0}u,v>& ={\int }_{\Omega }{\displaystyle \frac{u-u_0}{h}}vdx,\hfill \\ \hfill <A^{1}u,v>& ={\int }_{\Omega }\omega {|\nabla u|}^{p-2}\nabla u.\nabla vs.dx,\hfill \\ \hfill <A^{2}u,v>& ={\int }_{\Omega }\omega {\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\nabla u·\nabla vdx,\hfill \\ \hfill <A^{3}u,v>& ={\int }_{\Omega }\omega {\left|u\right|}^{p-2}uvdx,\hfill \\ \hfill <A^{4}u,v>& ={\int }_{\Omega }{\left[f\right]}_h(x,0)vdx.\hfill \end{array}$$

• Assertion 1. We claim that the operator A is monotone.

First, noting that

$$⟨A^{0}u-A^{0}v,u-v⟩={\int }_{\Omega }{\displaystyle \frac{{(u-v)}^2}{h}}\ge 0.$$

(9)

Moreover, by Lemma (1) and (2), the quantities

$$\begin{array}{c}\hfill ⟨A^{1}u-A^{1}v,u-v⟩\ge 0,\end{array}$$

(10)

$$\begin{array}{c}\hfill ⟨A^{2}u-A^{2}v,u-v⟩\ge 0.\end{array}$$

(11)

On the other hand, by the monotonicity of the mapping $\lambda \to {\left|\lambda \right|}^{p-2}\lambda$, we obtain, for $\lambda \in \mathbb{R}$,

$$\begin{array}{cc}\hfill & ⟨A^{3}u-A^{3}v,u-v⟩={\int }_{\Omega }\omega \left({\left|u\right|}^{p-2}u-{\left|v\right|}^{p-2}v\right)(u-v)dx\ge 0.\hfill \end{array}$$

(12)

Then, we deduce from (9), (10), (11), and (12) that

$$\begin{array}{c}\hfill ⟨Au-Av,u-v⟩\ge 0.\end{array}$$

Hence, the operator A is monotone.

• Assertion 2. We claim that the operator A is coercive.

We have

$$⟨A^{0}u,u⟩={\int }_{\Omega }{\displaystyle \frac{(u-u_0)u}{h}}\ge {\displaystyle \frac{1}{2}}{\int }_{\Omega }{\displaystyle \frac{u^2}{h}}-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\displaystyle \frac{u_0^2}{h}}={\displaystyle \frac{1}{2h}}{\parallel u\parallel }_{L^2(\Omega )}^2-{\displaystyle \frac{1}{2h}}{\parallel u_0\parallel }_{L^2(\Omega )}^2.$$

Using Hölder’s inequality, and Proposition 1, we obtain

$$\begin{array}{cc}\hfill <A^{4}u,u>={\int }_{\Omega }{\left[f\right]}_h(x,0)u& ={\int }_{\Omega }{\omega }^{1/p^{\prime }}{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}{\omega }^{1/p}udx\hfill \\ \hfill & \le {\left({\int }_{\Omega }\omega {\left|{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}\right|}^{p^{\prime }}\right)}^{1/p^{\prime }}{\left({\int }_{\Omega }\omega {\left|u\right|}^{p}dx\right)}^{1/p}\hfill \\ \hfill & \le C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}{∥u∥}_{W_0^{1,p}(\Omega ,\omega )}\hfill \\ \hfill & \le C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}{∥u∥}_E.\hfill \end{array}$$

(13)

On the other hand, it is obvious that

$$<A^{2}u,u>+<A^{3}u,u>={\int }_{\Omega }\omega {\displaystyle \frac{{|\nabla u|}^{2p}}{\sqrt{1+{|\nabla u|}^{2p}}}}dx+{\int }_{\Omega }\omega {\left|u\right|}^{p}dx\ge 0,$$

which implies that

$$\begin{array}{cc}\hfill & <Au,u>\hfill \\ & =<A^{0}u,u>+<A^{1}u,u>+<A^{2}u,u>+<A^{3}u,u>-<A^{4}u,u>\hfill \\ \hfill & \ge {\displaystyle \frac{1}{2h}}{\parallel u\parallel }_{L^2(\Omega )}-{\displaystyle \frac{1}{2h}}\parallel u_0{\parallel }_{L^2(\Omega )}+{\parallel u\parallel }_{W_0^{1,p}(\Omega ,\omega )}^p-C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,{\omega }_1)}{∥u∥}_E\hfill \\ & \ge min\left({\displaystyle \frac{1}{2h}},1\right)\left({\parallel u\parallel }_{L^2(\Omega )}+{\parallel u\parallel }_{W_0^{1,p}(\Omega ,\omega )}^p\right)-{\displaystyle \frac{1}{2h}}{\parallel u_0\parallel }_{L^2(\Omega )}-C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}{∥u∥}_E\hfill \\ \hfill & =min\left({\displaystyle \frac{1}{2h}},1\right){\parallel u\parallel }_E^p-C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}{∥u∥}_E.\hfill \end{array}$$

Therefore

$$\begin{array}{c}\hfill {\displaystyle \frac{<Au,u>}{{\parallel u\parallel }_E}}\to +\infty \mathrm{as}{\parallel u\parallel }_E\to +\infty .\end{array}$$

(14)

Thus, the operator A is coercive.

• Assertion 3. Finally, we claim that the operator A is bounded.

By Hölder’s inequality,

$$|<A^{0}u,v>|\le {\int }_{\Omega }{\displaystyle \frac{|u-u_0|}{h}}\left|v\right|dx\le {\displaystyle \frac{1}{h}}\parallel u-u_0{\parallel }_{L^2(\Omega )}{\parallel v\parallel }_{L^2(\Omega )}.$$

(15)

Applying Hölder’s inequality once more, we obtain

$$\begin{array}{cc}\hfill |<A^{1}u,v>|& =\left|{\int }_{\Omega }\omega {|\nabla u|}^{p-2}\nabla u.\nabla vs.dx\right|\hfill \\ \hfill & \le {\int }_{\Omega }{\omega }^{1/p^{\prime }}{|\nabla u|}^{p-1}{\omega }^{1/p}|\nabla v|dx\hfill \\ \hfill & \le {\left({\int }_{\Omega }\omega {|\nabla u|}^{p}dx\right)}^{1/p^{\prime }}{\left({\int }_{\Omega }\omega {|\nabla v|}^{p}dx\right)}^{1/p}\hfill \\ \hfill & \le {\left|\right|\nabla u\left|\right|}_{L^p(\Omega ,\omega )}^{p/p^{\prime }}{\left|\right|v\left|\right|}_{W_0^{1,p}(\Omega ,\omega )}\hfill \\ \hfill & \le {\left|\right|\nabla u\left|\right|}_{L^p(\Omega ,\omega )}^{p/p^{\prime }}{\left|\right|v\left|\right|}_E,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill |<A^{2}u,v>|& =\mid {\int }_{\Omega }\omega {\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\nabla u.\nabla vdx|\hfill \\ \hfill & \le {\int }_{\Omega }\omega {\displaystyle \frac{{|\nabla u|}^p}{\sqrt{1+{|\nabla u|}^{2p}}}}{|\nabla u|}^{p-2}|\nabla u.\nabla v|dx\hfill \\ \hfill & \le {\int }_{\Omega }\omega {\displaystyle \frac{{|\nabla u|}^p}{\sqrt{1+{|\nabla u|}^{2p}}}}{|\nabla u|}^{p-1}|\nabla v|dx\hfill \\ \hfill & \le {\int }_{\Omega }{\omega |\nabla u|}^{p-1}|\nabla v|dx\hfill \\ \hfill & \le {\left|\right|\nabla u\left|\right|}_{L^p(\Omega ,\omega )}^{p/p^{\prime }}{\left|\right|v\left|\right|}_{W_0^{1,p}(\Omega ,\omega )}\hfill \\ \hfill & \le {\left|\right|\nabla u\left|\right|}_{L^p(\Omega ,\omega )}^{p/p^{\prime }}{\left|\right|v\left|\right|}_E.\hfill \end{array}$$

(17)

Similarly, by applying Hölder’s inequality once again, together with Proposition 1, we have

$$\begin{array}{cc}\hfill & \left|<A^{3}u,v>-<A^{4}u,v>\right|\hfill \\ \hfill & =\left|{\int }_{\Omega }\omega {\left|u\right|}^{p-2}uvdx-{\int }_{\Omega }{\left[f\right]}_h(x,0)vdx\right|\hfill \\ \hfill & \le {\int }_{\Omega }{\omega }^{1/p^{\prime }}{\left|u\right|}^{p-1}{\omega }^{1/p}\left|v\right|dx+{\int }_{\Omega }{\omega }^{1/p^{\prime }}\left|{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}\right|{\omega }^{1/p}\left|v\right|dx\hfill \\ \hfill & \le {\left({\int }_{\Omega }\omega {\left|u\right|}^{p}dx\right)}^{1/p^{\prime }}{\left({\int }_{\Omega }\omega {|vs.|}^{p}dx\right)}^{1/p}+{\left({\int }_{\Omega }\omega {\left|{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}\right|}^{p^{\prime }}dx\right)}^{1/p^{\prime }}{\left({\int }_{\Omega }\omega {\left|v\right|}^{p}dx\right)}^{1/p}\hfill \\ \hfill & \le {C\parallel u\parallel }_{L^p(\Omega ,\omega )}^{p/p^{\prime }}{\parallel v\parallel }_{W_0^{1,p}(\Omega ,\omega )}+C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}{∥v∥}_{W_0^{1,p}(\Omega ,\omega )}\hfill \\ \hfill & \le {C\parallel u\parallel }_{L^p(\Omega ,\omega )}^{p/p^{\prime }}{\parallel v\parallel }_E+C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}{∥v∥}_E.\hfill \end{array}$$

(18)

Combining (15)–(18), we derive

$$\begin{array}{cc}\hfill & |<Au,v>|\hfill \\ & \le |<A^{0}u,v>|+|<A^{1}u,v>|+|<A^{2}u,v>|+|<A^{3}u,v>-<A^{4}u,v>|\hfill \\ \hfill & \le \left[{\displaystyle \frac{1}{h}}{\left|\right|u\left|\right|}_{L^{p^{\prime }}(\Omega )}^{1/p^{\prime }}+{2\left|\right|\nabla u\left|\right|}_{L^p(\Omega ,\omega )}^{p/p^{\prime }}+C{\parallel u\parallel }_{L^p(\Omega ,\omega )}^{p/p^{\prime }}+C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}\right]{\parallel v\parallel }_E\hfill \\ \hfill & \le C_1{∥v∥}_E,\hfill \end{array}$$

(19)

where $C_1:={\displaystyle \frac{1}{h}}{\left|\right|u\left|\right|}_{L^{p^{\prime }}(\Omega )}^{1/p^{\prime }}+{2\left|\right|\nabla u\left|\right|}_{L^p(\Omega ,\omega )}^{p/p^{\prime }}+C{\parallel u\parallel }_{L^p(\Omega ,\omega )}^{p/p^{\prime }}+C{∥{\displaystyle \frac{{\left[f\right]}_h(x,0)}{\omega }}∥}_{L^{p^{\prime }}(\Omega ,\omega )}.$

Therefore, A is bounded.

Furthermore, the operator A exhibits hemicontinuity. By Theorem 1, we conclude that the problem (7) has weak solutions.

Uniqueness of a weak solutions:

We shall now demonstrate that the problem (7) possesses a unique weak solution. We assume the existence of two weak solutions, $u_1$ and $u_2$, for problem (7). Consequently, we possess

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\displaystyle \frac{u_1-u_2}{h}}\varphi dx+{\int }_{\Omega }\omega \left(|\nabla u_1{|}^{p-2}\nabla u_1-{|\nabla u_2|}^{p-2}\nabla u_2\right)·\nabla \varphi dx\hfill \\ \hfill & +{\int }_{\Omega }\omega \left({\displaystyle \frac{|\nabla u_1{|}^{2p-2}}{\sqrt{1+|\nabla u_1{|}^{2p}}}}\nabla u_1-{\displaystyle \frac{|\nabla u_2{|}^{2p-2}}{\sqrt{1+|\nabla u_2{|}^{2p}}}}\nabla u_2\right)·\nabla \varphi dx\hfill \\ \hfill & +{\int }_{\Omega }\omega \left|\right(u_1{|}^{p-2}{u}_1-|u_2{|}^{p-2}{u}_2)\varphi dx\hfill \\ & =0.\hfill \end{array}$$

(20)

By substituting $\varphi =u_1-u_2$ into (20), we obtain

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\displaystyle \frac{{(u_1-u_2)}^2}{h}}dx+{\int }_{\Omega }\omega \left(|\nabla u_1{|}^{p-2}\nabla u_1-{|\nabla u_2|}^{p-2}\nabla u_2\right)·\nabla (u_1-u_2)dx\hfill \\ \hfill & +{\int }_{\Omega }\omega \left({\displaystyle \frac{|\nabla u_1{|}^{2p-2}}{\sqrt{1+|\nabla u_1{|}^{2p}}}}\nabla u_1-{\displaystyle \frac{|\nabla u_2{|}^{2p-2}}{\sqrt{1+|\nabla u_2{|}^{2p}}}}\nabla u_2\right)·\nabla (u_1-u_2)dx\hfill \\ \hfill & +{\int }_{\Omega }\omega \left(\right|u_1{|}^{p-2}{u}_1-|u_2{|}^{p-2}{u}_2)(u_1-u_2)dx\hfill \\ \hfill & =0.\hfill \end{array}$$

By Lemmas 1 and 2, we have

$$\begin{array}{cc}\hfill & {\int }_{\Omega }\omega \left(|\nabla u_1{|}^{p-2}\nabla u_1-{|\nabla u_2|}^{p-2}\nabla u_2\right)·\nabla (u_1-u_2)dx\ge 0,\hfill \\ & {\int }_{\Omega }\omega \left({\displaystyle \frac{|\nabla u_1{|}^{2p-2}}{\sqrt{1+|\nabla u_1{|}^{2p}}}}\nabla u_1-{\displaystyle \frac{|\nabla u_2{|}^{2p-2}}{\sqrt{1+|\nabla u_2{|}^{2p}}}}\nabla u_2\right)·\nabla (u_1-u_2)dx\ge 0,\hfill \\ & {\int }_{\Omega }\omega \left(\right|u_1{|}^{p-2}{u}_1-|u_2{|}^{p-2}{u}_2)(u_1-u_2)dx\ge 0.\hfill \end{array}$$

Therefore, ${\int }_{\Omega }{\displaystyle \frac{{(u_1-u_2)}^2}{h}}=0$. Hence $u_1=u_2$ a.e. in $\Omega .$ □

Now we are ready to state the main result for this section.

Definition 2.

A function u is called a weak solution of the problem (4) if $u\in W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega )$ and

$$\begin{array}{c}{\int }_{\Omega }{\displaystyle \frac{u_k-u_{k-1}}{h}}\varphi dx+{\int }_{\Omega }\omega \left(|\nabla u_k{|}^{p-2}+{\displaystyle \frac{|\nabla u_k{|}^{2p-2}}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right)\nabla u_k·\nabla \varphi dx+{\int }_{\Omega }\omega {\left|u_k\right|}^{p-2}{u}_k\varphi dx\hfill \\ ={\int }_{\Omega }{\left[f\right]}_h(x,(k-1)h)\varphi dx\hfill \end{array}$$

(21)

for all $\varphi \in W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega ).$

Theorem 3.

Let $\left({\left[f\right]}_h(·,(k-1)h)/\omega \right)\in L^{p^{\prime }}(\Omega ,\omega )$, and assume that hypotheses $\left(H_1\right)$ and $\left(H_2\right)$ hold. Then problem (4) has a unique weak solution.

Proof. 

Let $k=1$, from Theorem 2, there exists a weak solution $u_1\in W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega )$. By induction, we deduce in the same manner that for k, the problem (4) has a unique weak solution $u_k\in W_0^{1,p}(\Omega ,\omega )\cap L^2(\Omega )$. □

4. Existence and Uniqueness of Weak Solutions for Problem (1)

This section investigates the existence and uniqueness of the weak solution of (1).

Definition 3.

We say that $u\in L^p(0,T;W^{1,p}(\Omega ,\omega ))\cap C(0,T;L^2(\Omega ))$ such that ${\displaystyle \frac{\partial u}{\partial t}}\in L^{p^{\prime }}(0,T;{\left(W_0^{1,p}(\Omega ,\omega )\right)}^{*})$ is a weak solution of the problem (1) if

$$\begin{array}{cc}\hfill & {\int }_Q{\displaystyle \frac{\partial u}{\partial t}}\varphi dxdt+{\int }_Q\omega \left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u·\nabla \varphi dxdt+{\int }_Q\omega {\left|u\right|}^{p-2}u\varphi dxdt\hfill \\ \hfill & ={\int }_{Q}f\varphi dxdt\hfill \end{array}$$

for all $\varphi \in L^p(0,T;W_0^{1,p}(\Omega ,\omega )).$

Now we are ready to state the main result of this work.

Theorem 4.

Let $f/\omega \in L^{p^{\prime }}(Q,\omega )$ and assume that hypotheses $\left(H_1\right)$ and $\left(H_2\right)$ hold. Then problem (1) has a unique weak solution.

Proof. 

Let n be a positive integer and let $h={\displaystyle \frac{T}{n}}.$ We define

$$u_h(x,t):=\left\{\begin{array}{cc}{u}_0\left(x\right),\hfill & t=0\hfill \\ u_1\left(x\right),\hfill & 0<t\le h\hfill \\ ⋮\hfill \\ u_n\left(x\right),\hfill & (n-1)h<t\le T\hfill \end{array}\right.$$

(22)

where $u_k$ (for $k=1,\dots ,n$) is the weak solution of the problem (4).

Choose $u_k$ as a test function in the weak formulation of problem (4), which we obtain by Hölder’s and Young’s inequalities

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\displaystyle \frac{u_k^2}{h}}dx+{\int }_{\Omega }\omega \left(1+{\displaystyle \frac{|\nabla u_k{|}^p}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right)|\nabla u_k{|}^{p}dx+{\int }_{\Omega }\omega {\left|u_k\right|}^{p}dx\hfill \\ \hfill & ={\int }_{\Omega }{\left[f\right]}_h(x,(k-1)h)u_{k}dx+{\int }_{\Omega }{\displaystyle \frac{u_k{u}_{k-1}}{h}}dx\hfill \\ \hfill & \le {\int }_{\Omega }{\left[f\right]}_h(x,(k-1)h)u_{k}dx+{\int }_{\Omega }{\displaystyle \frac{|u_k{|}^2}{2h}}dx+{\int }_{\Omega }{\displaystyle \frac{|u_{k-1}{|}^2}{2h}}dx\hfill \\ \hfill & ={\int }_{\Omega }{w}^{1/p^{\prime }}{\displaystyle \frac{{\left[f\right]}_h(x,(k-1)h)}{w}}{w}^{1/p}{u}_{k}dx+{\int }_{\Omega }{\displaystyle \frac{|u_k{|}^2}{2h}}dx+{\int }_{\Omega }{\displaystyle \frac{|u_{k-1}{|}^2}{2h}}dx\hfill \\ & \le {∥{\displaystyle \frac{{\left[f\right]}_h(x,(k-1)h)}{w}}∥}_{L^{p^{\prime }}(\Omega ,w)}{\parallel u_k\parallel }_{L^p(\Omega ,w)}+{\int }_{\Omega }{\displaystyle \frac{|u_k{|}^2}{2h}}dx+{\int }_{\Omega }{\displaystyle \frac{|u_{k-1}{|}^2}{2h}}dx\hfill \\ & \le {\displaystyle \frac{1}{p^{\prime }}}{∥{\displaystyle \frac{{\left[f\right]}_h(x,(k-1)h)}{w}}∥}_{L^{p^{\prime }}(\Omega ,w)}^{p^{\prime }}+{\displaystyle \frac{1}{p}}{|u_k\parallel }_{L^p(\Omega ,w)}^p+{\int }_{\Omega }{\displaystyle \frac{|u_k{|}^2}{2h}}dx+{\int }_{\Omega }{\displaystyle \frac{|u_{k-1}{|}^2}{2h}}dx\hfill \\ & \le {∥{\displaystyle \frac{{\left[f\right]}_h(x,(k-1)h))}{w}}∥}_{L^{p^{\prime }}(\Omega ,w)}^{p^{\prime }}+{\parallel u_k\parallel }_{L^p(\Omega ,w)}^p+{\int }_{\Omega }{\displaystyle \frac{|u_k{|}^2}{2h}}dx+{\int }_{\Omega }{\displaystyle \frac{|u_{k-1}{|}^2}{2h}}dx.\hfill \end{array}$$

(23)

Hence

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel u_k{\parallel }_{L^2(\Omega )}^2}{2h}}+{\int }_{\Omega }\omega \left(1+{\displaystyle \frac{|\nabla u_k{|}^p}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right){|\nabla u_k|}^{p}dx\hfill \\ & \le {∥{\displaystyle \frac{{\left[f\right]}_h(x,(k-1)h)}{w}}∥}_{L^{p^{\prime }}(\Omega ,\omega )}^{p^{\prime }}+{\displaystyle \frac{\parallel u_{k-1}{\parallel }_{L^2(\Omega )}^2}{2h}}.\hfill \end{array}$$

(24)

For each $t\in (0,T)$, there exists $j\in \{0,\dots ,n\}$ such that $t\in \left(\right(j-1)h,jh)$. Then, by adding inequality (24) from $k=1$ to $k=j$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel u_j{\parallel }_{L^2(\Omega )}^2}{2}}+h\sum _{k=1}^j{\int }_{\Omega }\omega \left(1+{\displaystyle \frac{|\nabla u_k{|}^p}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right){|\nabla u_k|}^{p}dx\hfill \\ \hfill & \le h\sum _{k=1}^j{∥{\displaystyle \frac{{\left[f\right]}_h(x,(k-1)h)}{w}}∥}_{L^{p^{\prime }}(\Omega ,\omega )}+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2(\Omega )}^2}{2}}.\hfill \end{array}$$

(25)

Now by the expression (22) of $u_h$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel u_h{\left(t\right)\parallel }_{L^2(\Omega )}^2}{2}}+{\int }_0^{jh}{\int }_{\Omega }\omega \left(1+{\displaystyle \frac{|\nabla u_h{|}^p}{\sqrt{1+|\nabla u_h{|}^{2p}}}}\right){|\nabla u_h|}^{p}dxds\hfill \\ & \le {\int }_0^{jh}{\parallel f/w\parallel }_{L^{p^{\prime }}(\Omega ,\omega )}ds+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2}^2(\Omega )}{2}}.\hfill \end{array}$$

In particular, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel u_h{\left(t\right)\parallel }_{L^2(\Omega )}^2}{2}}+{\int }_0^t{\int }_{\Omega }\omega \left(1+{\displaystyle \frac{|\nabla u_h{|}^p}{\sqrt{1+|\nabla u_h{|}^{2p}}}}\right){|\nabla u_h|}^{p}dxds\hfill \\ \hfill & \le {\int }_0^T{\parallel f/w\parallel }_{L^{p^{\prime }}(\Omega ,\omega )}ds+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2(\Omega )}^2}{2}}.\hfill \end{array}$$

(26)

By taking the supermum over $[0,T]$, we obtain

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\displaystyle \frac{\parallel u_h{\left(t\right)\parallel }_{L^2(\Omega )}^2}{2}}+{\int }_0^t{\int }_{\Omega }\omega \left(1+{\displaystyle \frac{|\nabla u_h{|}^p}{\sqrt{1+|\nabla u_h{|}^{2p}}}}\right){|\nabla u_h|}^{p}dxds\hfill \\ & \le {\int }_0^T{\parallel f/w\parallel }_{L^{p^{\prime }}(\Omega ,\omega )}ds+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2(\Omega )}^2}{2}}.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill u_h& \rightharpoonup u{\mathrm{weakly}}^{*}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega )),\hfill \\ \hfill u_h& \rightharpoonup u\mathrm{weakly}\mathrm{in}{L}^p(0,T;L^p(\Omega ,\omega )),\hfill \\ \hfill \left[|\nabla u_h{|}^{p-2}+{\displaystyle \frac{|\nabla u_h{|}^{2p-2}}{\sqrt{1+|\nabla u_h{|}^{2p}}}}\right]\nabla u_h& \rightharpoonup \alpha \mathrm{weakly}\mathrm{in}{L}^{p^{\prime }}(0,T;L^{p^{\prime }}(\Omega ,{\omega }^{1-p^{\prime }})),\hfill \end{array}$$

where $\alpha$ will be determined later.

Next, we prove that u is a weak solution of problem (1.1). For each $\varphi \in C^1\left(\overline{Q}\right)$ with $\varphi (0,T)=0$, we take $\varphi (x,kh)$ as a test function in (21). By summing the equalities (21) from $k=1,\dots ,n$, we have

$$\begin{array}{cc}\hfill & \sum _{k=0}^{n-1}{\int }_{\Omega }{u}_k(\varphi (x,kh)-(\varphi (x,(k+1)h)dx-{\int }_{\Omega }{u}_0\varphi (x,0)dx\hfill \\ & +\sum _{k=0}^n{\int }_{\Omega }\omega \left(|\nabla u_k{|}^{p-2}+{\displaystyle \frac{|\nabla u_k{|}^{2p-2}}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right)\nabla u_k·\nabla \varphi (x,kh)dx\hfill \\ & +\sum _{k=0}^n{\int }_{\Omega }\omega {\left|u_k\right|}^{p-2}{u}_k\varphi (x,kh)dx\hfill \\ \hfill & =h\sum _{k=0}^n{\int }_{\Omega }{\left[f\right]}_h(x,(k-1)h)(\varphi (x,kh)dx.\hfill \end{array}$$

(27)

In view of the definition of $u_h(x,t)$ in (22), we obtain

$$\begin{array}{cc}\hfill \sum _{k=0}^{n-1}{\int }_{\Omega }{u}_k(\varphi (x,kh)-(\varphi (x,(k+1)h)dx& =-\sum _{k=0}^{n-1}{\int }_{kh}^{(k+1)h}{\int }_{\Omega }{u}_h\left(x\right){\displaystyle \frac{\partial \varphi (x,t)}{\partial t}}dxdt\hfill \\ & =-{\int }_0^T{\int }_{\Omega }{u}_h(x,t){\displaystyle \frac{\partial \varphi (x,t)}{\partial t}}dxdt\hfill \\ & \to -{\int }_{Q}u(x,t){\displaystyle \frac{\partial \varphi (x,t)}{\partial t}}dxdt,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \sum _{k=0}^n{\int }_{\Omega }\omega \left(|\nabla u_k{|}^{p-2}+{\displaystyle \frac{|\nabla u_k{|}^{2p-2}}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right)\nabla u_k·\nabla \left(\varphi (x,kh)\right)dx\hfill \\ & ={\int }_0^T{\int }_{\Omega }\omega \left(|\nabla u_h{|}^{p-2}+{\displaystyle \frac{|\nabla u_h{|}^{2p-2}}{\sqrt{1+|\nabla u_h{|}^{2p}}}}\right)\nabla u_h·\nabla \left(\varphi (x,t)\right)dxdt\hfill \\ & +\sum _{k=1}^n{\int }_{(k-1)h}^{kh}{\int }_{\Omega }\omega \left(|\nabla u_k{|}^{p-2}+{\displaystyle \frac{|\nabla u_k{|}^{2p-2}}{\sqrt{1+|\nabla u_k{|}^{2p}}}}\right)\nabla u_k·(\nabla (\varphi (x,kh)-\nabla \varphi (x,t))dxdt\hfill \\ & \to {\int }_Q\omega \alpha .\nabla \varphi (x,t)dxdt.\hfill \end{array}$$

Then, combining all these relations and taking $h\to 0$ in (27), we deduce

$$\begin{array}{cc}\hfill & -{\int }_{Q}u(x,t){\displaystyle \frac{\partial \varphi (x,t)}{\partial t}}dxdt-{\int }_{\Omega }{u}_0\varphi (x,0)dx+{\int }_Q\omega \alpha ·\nabla \varphi dxdt+{\int }_Q\omega {\left|u\right|}^{p-2}u\varphi dxdt\hfill \\ & ={\int }_{Q}f\varphi dxdt\hfill \end{array}$$

(28)

Moreover, if $\varphi \in C_c^{\infty }(\Omega )$ in (28), we obtain

$$\begin{array}{cc}\hfill & -{\int }_{Q}u\left(x\right){\displaystyle \frac{\partial \varphi (x,t)}{\partial t}}dxdt+{\int }_Q\omega \alpha ·\nabla \varphi dxdt+{\int }_Q\omega {\left|u\right|}^{p-2}u\varphi dxdt={\int }_{Q}f\varphi dxdt.\hfill \end{array}$$

(29)

Next we prove that $\alpha =\left[{|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right]\nabla u$.

Let $Bu:=\left[{|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right]\nabla u$. Note that by summing inequalities (23) for $k=1,\dots ,n$, we have

$${\displaystyle \frac{\parallel u_h{\left(T\right)\parallel }_{L^2(\Omega )}^2}{2}}+{\int }_Q\omega B{u}_h.\nabla u_{h}dxdt+{\int }_Q\omega {\left|u_h\right|}^{p}dxdt\le {\int }_{Q}f{u}_{h}dxdt+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2(\Omega )}^2}{2}}.$$

Observe that

$$\begin{array}{cc}\hfill & B{u}_h.\nabla u_h=(B{u}_h-Bv).(\nabla u_h-\nabla v)+B{u}_h.\nabla v+Bv.\nabla u_h-Bv\nabla v,\hfill \end{array}$$

then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel u_h{\left(T\right)\parallel }_{L^2(\Omega )}^2}{2}}+{\int }_Q\omega [(B{u}_h-Bv).(\nabla u_h-\nabla v)+B{u}_h.\nabla v+Bv.\nabla u_h-Bv\nabla v]dxds\hfill \\ & +{\int }_Q\omega {\left|u_h\right|}^{p}dxdt\le {\int }_{Q}f{u}_{h}dxdt+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2(\Omega )}^2}{2}}.\hfill \end{array}$$

Recall, from (10) and (11), that

$${\int }_Q\omega (B{u}_h-Bv).(\nabla u_h-\nabla v)dxdt\ge 0.$$

Therefore,

$$\begin{array}{c}{\displaystyle \frac{\parallel u_h{\left(T\right)\parallel }_{L^2(\Omega )}^2}{2}}+{\int }_Q\omega (B{u}_h.\nabla v+Bv.\nabla u_h-Bv\nabla vs.)dxds+{\int }_Q\omega {\left|u_h\right|}^{p}dxds\hfill \\ \le {\int }_{Q}f{u}_{h}dxdt{\displaystyle \frac{\parallel u_0{\parallel }_{L^2}^{(\Omega )}}{2}}\hfill \end{array}$$

(30)

Passing $h\to 0$ in (30) and letting $\varphi =u$ in (8), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\parallel u\left(T\right)\parallel }_{L^2}^2}{2}}+{\int }_Q\omega (\alpha .\nabla v+Bv.\nabla u-Bv\nabla vs.)dxds+{\int }_Q\omega {\left|u\right|}^{p}dxds\hfill \\ & \le {\int }_{Q}fudxdt+{\displaystyle \frac{\parallel u_0{\parallel }_{L^2}^2}{2}}\hfill \\ & =-{\displaystyle \frac{1}{2}}{\int }_{\Omega }{\left|u\left(T\right)\right|}^{2}dx+{\int }_Q\omega \alpha ·\nabla udxdt+{\int }_Q\omega {\left|u\right|}^{p}dxdt.\hfill \end{array}$$

Hence

$$\begin{array}{cc}\hfill & {\int }_Q\omega (\alpha .\nabla v+Bv.\nabla u-Bv.\nabla v)dxdt\le {\int }_Q\omega \alpha ·\nabla udxdt.\hfill \end{array}$$

(31)

That leads to

$$\begin{array}{c}\hfill {\int }_Q\omega \left(\alpha -Bv\right).(\nabla v-\nabla u)dxdt\le 0.\end{array}$$

(32)

If we choose $v=u-\lambda \psi$ for any $\lambda >0$ and $\varphi \in L^p(0,T;L^p(\Omega ,\omega ))$, we obtain

$$\begin{array}{cc}\hfill & {\int }_Q\omega \left(\alpha -\left[{|\nabla (u-\lambda \psi )|}^{p-2}+{\displaystyle \frac{{|\nabla (u-\lambda \psi )|}^{2p-2}}{\sqrt{1+{|\nabla (u-\lambda \psi )|}^{2p}}}}\right]\nabla (u-\lambda \psi )\right).\nabla \psi dxdt\ge 0.\hfill \end{array}$$

(33)

Hence, by sending $\lambda \to 0$, we deduce

$${\int }_Q\omega \left(\alpha -\left[{|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right]\nabla u\right).\nabla \psi dxdt=0.$$

Therefore $\alpha =\left[{\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right]\nabla u$ a.e. on Q. Moreover, the equality (29) becomes

$$\begin{array}{cc}\hfill & -{\int }_{Q}u(x,t){\displaystyle \frac{\partial \varphi (x,t)}{\partial t}}dxdt+{\int }_Q\omega \left[{\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right]·\nabla \varphi dxdt\hfill \\ & +{\int }_Q\omega {\left|u\right|}^{p-2}u\varphi dxdt={\int }_{Q}f\varphi dxdt.\hfill \end{array}$$

(34)

Recalling the fact that $u\in L^p\left(0,T;W_0^{1,p}(\Omega ,\omega )\right)$ and ${\displaystyle \frac{\partial u}{\partial t}}\in L^{p^{\prime }}(0,T;{\left(W_0^{1,p}(\Omega ,\omega )\right)}^{\prime })$ from the above equation, we conclude that u belongs to $C\left([0,T];L^2(\Omega )\right)$, which completes our proof of the existence of the weak solutions for problem (1).

Now, we show that this weak solution is unique. Suppose there exist two weak solutions u and v for the problem (1). By (34), we have

$$\begin{array}{cc}\hfill & {\int }_Q{\displaystyle \frac{\partial (u-v)}{\partial t}}\varphi dxdt\hfill \\ & +{\int }_Q\omega \left[\left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u-\left({|\nabla v|}^{p-2}+{\displaystyle \frac{{|\nabla v|}^{2p-2}}{\sqrt{1+{|\nabla v|}^{2p}}}}\right)\nabla v\right]·\nabla \varphi dxdt\hfill \\ & +{\int }_Q{\omega \left(\right|u|}^{p-2}u-{\left|v\right|}^{p-2}v)\varphi dxdt=0.\hfill \end{array}$$

(35)

If we choose $u-v$ as the test function in (35), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}{\int }_{\Omega }{(u\left(T\right)-v\left(T\right))}^{2}dx\hfill \\ & +{\int }_Q\omega \left[\left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u-\left({|\nabla v|}^{p-2}+{\displaystyle \frac{{|\nabla v|}^{2p-2}}{\sqrt{1+{|\nabla v|}^{2p}}}}\right)\nabla v\right]·\nabla (u-v)dxdt\hfill \\ & +{\int }_Q{\omega \left(\right|u|}^{p-2}u-{\left|v\right|}^{p-2}v)(u-v)dxdt=0.\hfill \end{array}$$

Noting, from Lemmas 1 and 2, that

$$\begin{array}{cc}\hfill & {\int }_Q\omega \left[\left({|\nabla u|}^{p-2}+{\displaystyle \frac{{|\nabla u|}^{2p-2}}{\sqrt{1+{|\nabla u|}^{2p}}}}\right)\nabla u-\left({|\nabla v|}^{p-2}+{\displaystyle \frac{{|\nabla v|}^{2p-2}}{\sqrt{1+{|\nabla v|}^{2p}}}}\right)\nabla v\right]·\nabla (u-v)dxdt\ge 0,\hfill \\ & {\int }_Q{\omega \left(\right|u|}^{p-2}u-{\left|v\right|}^{p-2}v)(u-v)dxdt\ge 0.\hfill \end{array}$$

Then ${\int }_{\Omega }{(u\left(T\right)-v\left(T\right))}^{2}dx=0$. Hence $u=v$ a.e. on Q, which completes our proof. □

5. Conclusions

In this paper, we proved the existence and uniqueness of weak solutions for a class of nonlinear degenerate parabolic equations arising from capillarity and non-Newtonian fluid models under Dirichlet boundary conditions. The equation includes a weighted nonlinear diffusion operator that generalizes the classical p-Laplacian by combining spatial heterogeneity, nonlinear diffusion, and capillarity effects.

Using Rothe’s semi-discretization method in time, we reduced the parabolic problem to a sequence of nonlinear elliptic subproblems. The solvability of each discrete problem was established through monotone operator theory. By deriving suitable a priori estimates and performing a limiting process, we constructed a weak solution to the original parabolic problem and proved its uniqueness.

The results extend known elliptic capillarity-type equations to the time-dependent parabolic case with weighted coefficients. This work provides a rigorous mathematical framework for describing nonlinear diffusion and capillary-driven flow in heterogeneous media.