A Self-Similar Analysis of the Solutions to the Cross-Diffusion System

Abstract

We study the qualitative behaviour of weak solutions to a doubly nonlinear cross-diffusion system in an inhomogeneous medium with nonlinear boundary flux. The main novelty of the paper lies in the analysis of a cross-diffusive p-Laplacian system weighted by a spatially varying density in the form $\rho \left(x\right)={(1+|x\left|\right)}^n$, combined with nonlinear boundary interactions. By constructing self-similar weak solutions of Barenblatt type and employing a nonlinear separation of variables, the system is reduced to a coupled family of ordinary differential equations that characterise admissible similarity profiles. This approach allows us to identify Fujita-type critical conditions separating global solutions from finite-time blow-up and to derive explicit estimates for the solution profiles. Consequently, we establish critical thresholds driven by the interaction between cross-diffusion, degeneracy, boundary nonlinearity, and medium inhomogeneity.

1. Introduction

This paper investigates the qualitative properties of solutions for a doubly degenerate, non-Newtonian cross-diffusion system defined in a non-homogeneous medium with nonlinear boundary flux. In the area $Q=\left\{(x,t)\right|x\in {\mathbb{R}}_{+},t>0\},$ we consider a cross-diffusion system with nonlinear Neumann boundary conditions

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & \rho \left(x\right){\partial }_{t}u={\partial }_x\left(v^{m_1-1}{\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u\right)\hfill \\ \hfill & \rho \left(x\right){\partial }_{t}v={\partial }_x\left(u^{m_2-1}{\left|{\partial }_{x}v\right|}^{p-2}{\partial }_{x}v\right)\hfill \end{array}\right.,(x,t)\in Q;\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & -v^{m_1-1}{\left|{\partial }_{x}u\right|}^{p-2}{\partial }_{x}u{|}_{x=0}=u^{q_1}(0,t)\hfill \\ \hfill & -u^{m_2-1}{\left|{\partial }_{x}v\right|}^{p-2}{\partial }_{x}v{|}_{x=0}=v^{q_2}(0,t)\hfill \end{array}\right.,t>0;\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & u(x,0)=u_0\left(x\right)\hfill \\ \hfill & v(x,0)=v_0\left(x\right)\hfill \end{array}\right.,x\in {\mathbb{R}}_{+},\hfill \end{array}$$

(3)

where $\rho \left(x\right)={(1+|x\left|\right)}^n,m_i\ge 1,p>2,n,q_i>0,$ and the initial data $u_0,v_0$ are non-negative, bounded, compactly supported, with $u_0,v_0\in L^{\infty }\left({\mathbb{R}}_{+}\right)\cap W^{1,p}\left({\mathbb{R}}_{+}\right)$, and satisfies the Neumann condition at $t=0$:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & -v_0^{m_1-1}{\left|{\partial }_x{u}_0\right|}^{p-2}{\partial }_x{u}_0{|}_{x=0}=u_0^{q_1}\left(0\right),\hfill \\ \hfill & -u_0^{m_2-1}{\left|{\partial }_x{v}_0\right|}^{p-2}{\partial }_x{v}_0{|}_{x=0}=v_0^{q_2}\left(0\right).\hfill \end{array}\right.\end{array}$$

The present study is rooted in the classical theory of nonlinear parabolic equations, where the interplay between diffusion, nonlinearity, and boundary effects determines the long-time behaviour of solutions. A foundational result in this direction is due to Fujita [1], who showed that for the semilinear heat equation, the existence of global solutions or finite-time blow-up is governed by a critical exponent. This discovery initiated a vast body of work on critical phenomena in nonlinear diffusion. While classical models often rely on linear diffusion, many complex processes in physics [2], chemistry [3], biology [4], economy [5], and image processing [6,7] are governed by more intricate, degenerate, and non-local transport mechanisms [8,9]. A particularly significant area of research is cross-diffusion, where the diffusive flux of one component is coupled to the concentration of another [10,11]. Simultaneously, physical processes involving non-Newtonian fluids [12,13], heat conduction [14,15,16], or filtration in porous media [17,18] are often characterised by degenerate operators such as the p-Laplacian, which incorporates phenomena like finite propagation speeds and shear-dependent dynamics [19,20]. p-Laplacian diffusion generalises Fick’s law ($p=2$ case) to model non-Newtonian fluids [21] that obey the Ostwald-de Waele power law [22], and for $p>2$, the fluid exhibits shear-thickening behaviour [23]. Unlike the linear model, which has an infinite propagation speed of the diffusion or heat conductivity, p-Laplacian diffusion is characterised by a finite speed of propagation, leading to the formation of free boundaries or interfaces [24,25].

The cross-diffusion coupling reflects the fact that the effective diffusivity of one component depends explicitly on the concentration of the other. In such systems, the flux of species u is modulated by the presence of species v, and vice versa. This mechanism naturally arises in multicomponent filtration processes and in population dynamics, where interactions between species influence their mobility. A classic example is the Shigesada–Kawasaki–Teramoto model [26], which captures inter-species interference effects such as steric hindrance, volume exclusion, or repulsive taxis [27]. The density function $\rho \left(x\right)$ characterises the heterogeneity of the porous medium, representing its local porosity or specific storage capacity. This situation is typical in fractured porous media or in unbounded domains with expanding cross-sectional geometry, where the storage capacity is not uniform and significantly affects the temporal evolution of the transported quantities [25,28]. The condition (2) describes a nonlinear boundary flux induced by shear-dependent transport mechanisms. From a modelling perspective, it can be interpreted as a constitutive boundary law that relates the boundary flux to the trace of the solution at $x=0.$ In particular, the boundary condition introduces a nonlinear feedback mechanism at the interface, whereby the boundary exchange depends explicitly on the local state of the system [29].

For degenerate diffusion operators, such as the porous medium equation and the p-Laplacian, the situation is more intricate due to finite propagation speed, loss of uniform ellipticity, and the emergence of free boundaries. Comprehensive treatments of these phenomena can be found in the monographs by DiBenedetto [24] and Vázquez [17], which establish the functional approach, weak solution theory, and qualitative properties of degenerate parabolic equations. Boundary effects further enrich the dynamics. In particular, nonlinear Neumann boundary conditions may act as sources or sinks of mass or energy and can fundamentally alter the blow-up behaviour. Early systematic investigations of nonlinear boundary fluxes were carried out by Filo [30] and Galaktionov–Levine [31], who demonstrated that boundary nonlinearities give rise to Fujita-type critical exponents even in one-dimensional settings. J.Filo studied the nonlinear boundary flux problem for the diffusion equation [30], and he considered the following problem:

$$\begin{array}{cc}\hfill & {\partial }_t\left(\beta \left(u\right)\right)={\partial }_x^{2}u,(x,t)\in {\mathrm{\Omega }}_T=\{0<x<1,0<t<T\};\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & {\partial }_{x}u{|}_{x=0}=0,{\partial }_{x}u{|}_{x=1}=g\left(u(1,t)\right),t\in [0,T);\hfill \end{array}$$

$$\begin{array}{cc}\hfill & u(x,0)=u_0\left(x\right)\ge \delta >0,x\in [0,1],\hfill \end{array}$$

where $m>0,\beta \left(u\right)=u^m,g\left(u\right)=u^{\alpha },\alpha <\infty .$

J.Filo proved that if $u_0\in C^{2+\nu }\left([0,1]\right),$ for some $0<\nu <1,$ and satisfied initial-boundary conditions, then the problem (4) has a unique solution for a time $T^{*},0<T^{*}<\infty$, such that $u\in C^{2,1}(\overline{{\mathrm{\Omega }}_T)}.$ Moreover, he showed that

$$\underset{t\to T^{*}}{lim}{\parallel u(.,t)\parallel }_{\infty }=+\infty .$$

Note:

If $T^{*}<\infty$, then we say that the solution $u(x,t)$ blows up in a finite time; otherwise, the solution exists globally.

V.Galaktionov and H.Levini investigated the following kinds of problems [31]:

$$\begin{array}{cc}\hfill & {\partial }_{t}u={\partial }_x\left(\right|{\partial }_{x}u{|}^{m-1}{\partial }_{x}u),x\in {\mathbb{R}}_{+},t>0;\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & -|{\partial }_x{u|}^{m-1}{\partial }_{x}u{|}_{x=0}=u^p(0,t),t\ge 0;\hfill \end{array}$$

$$\begin{array}{cc}\hfill & u(x,0)=u_0\left(x\right)\ge 0,x>0,sup{u}_0<\infty ,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & sup|u_0^{\prime }|<\infty ,u_0\mathrm{is}\mathrm{compactly}\mathrm{supported},\mathrm{and}-\left|u_0^{\prime }\left(0\right)\right|=u_0^p\left(0\right),\hfill \end{array}$$

where $m>1.$

They showed that

-

If ${\displaystyle \frac{2m}{m+1}}<p\le p_c=2m,$ then the solutions of (5) blow up in a finite time for any nontrivial $u_0$;

-

If $p>p_c,$ then the solutions of (5) are global for sufficiently small $u_0$ and blow up in a finite time for sufficiently large $u_0$.

$p_c$ is also known as the Fujita critical exponent [1,32].

Later, J.Filo and M.Pérez-Llanos [33] extended these types of results and blow-up regions for this problem:

$$\begin{array}{cc}\hfill & {\partial }_t\left(u^m\right)={\partial }_x\left(\right|{\partial }_x{u|}^{r-1}{\partial }_{x}u),x\in 0<x<L,t\in 0<t<T\};\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\partial }_{x}u{|}_{x=0}=0,{\left|{\partial }_{x}u\right|}^{r-1}{\partial }_{x}u{|}_{x=L}=u^{\alpha }(L,t),t\in (0,T);\hfill \end{array}$$

$$\begin{array}{cc}\hfill & u(x,0)=u_0\left(x\right),x\in [0,L],\hfill \end{array}$$

where $m,r>0,\alpha <\infty .$

F.Quirós and J.D.Rossi considered the following system coupled through boundary flux [34]:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & {\partial }_{t}u={\partial }_x\left({\partial }_x{u}^m\right)\hfill \\ \hfill & {\partial }_{t}v={\partial }_x\left({\partial }_x{v}^n\right)\hfill \end{array}\right.,x\in {\mathbb{R}}_{+},t\in (0,T);\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & -{\partial }_x{u}^m=v^p\hfill \\ \hfill & -{\partial }_x{v}^n=u^q\hfill \end{array}\right.,x=0,t\in (0,T);\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & u(x,0)=u_0\left(x\right)\ge 0\hfill \\ \hfill & v(x,0)=v_0\left(x\right)\ge 0\hfill \end{array}\right.,x\in {\mathbb{R}}_{+},\hfill \end{array}$$

where $m,n>1,p,q>0.$

The authors showed the critical global existence and the critical Fujita-type curves:

(i)

${\left(pq\right)}_c={\displaystyle \frac{(m+1)(n+1)}{4}};$

(ii)

$min\{{\alpha }_i+{\beta }_i\}=0,$

where ${\alpha }_i={\displaystyle \frac{(2p+n)(2-i)+(2q+m)(i-1)+1}{(m+1)(n+1)-4pq}}$,  ${\beta }_i={\displaystyle \frac{\left(p\right(m-1)+m)(2-i)+\left(q\right(n-1)+n)(i-1)-2pq+mn}{(m+1)(n+1)-4pq}}$,  $i=1,2.$

Moreover, they demonstrated the following results:

(I)

If $pq\le {\left(pq\right)}_c,$ every non-negative solution of the problem (6) is global.

(II)

If $pq>{\left(pq\right)}_c,$ with $min\{{\alpha }_i+{\beta }_i\}\le 0,(i=1,2),$ then every non-negative, nontrivial blow-up occurs in a finite time.

(III)

If $pq>{\left(pq\right)}_c,$ with $max\{{\alpha }_i+{\beta }_i\}>0,(i=1,2),$ then the solution of the problem (6) blows up in a finite time for sufficiently large initial data, and globally exists for sufficiently small initial data.

S.Zheng et al. [35] extended these results to the following system:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & {\partial }_{t}u={\partial }_x\left({\partial }_x{u}^m\right)\hfill \\ \hfill & {\partial }_{t}v={\partial }_x\left({\partial }_x{v}^n\right)\hfill \end{array}\right.,x\in \mathbb{R},t\in [0,T);\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & -{\partial }_x{u}^m=u^{\alpha }{v}^p\hfill \\ \hfill & -{\partial }_x{v}^n=u^q{v}^{\beta }\hfill \end{array}\right.,x=0,t\in (0,T);\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill & u(x,0)=u_0\left(x\right)\ge 0\hfill \\ \hfill & v(x,0)=v_0\left(x\right)\ge 0\hfill \end{array}\right.,x\in \mathbb{R},\hfill \end{array}$$

and they obtained the critical curves as follows:

(i)

${\left(pq\right)}_c=\left({\displaystyle \frac{m+1}{2}}-\alpha \right)\left({\displaystyle \frac{n+1}{2}}-\beta \right);$

(ii)

$min\{l_i-k_i\}=0,$

where $\alpha ,\beta \ge 0,k_i={\displaystyle \frac{(2p-2\beta +n)(2-i)+(2q-2\alpha +m)(i-1)+1}{4pq-(m+1-2\alpha )(n+1-2\beta )}}$,  $l_i={\displaystyle \frac{1-k_i(m(2-i)+n(i-1)-1)}{2}}$,  $i=1,2.$

M.Aripov et al. [36] considered the following problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\partial }_t{u}_i={\partial }_x\left(|{\partial }_x{u}_i^k{|}^{m-1}{\partial }_x{u}_i^k\right)+u_i^{p_i},x\in {\mathbb{R}}_{+},t>0;\hfill \\ \hfill & -|{\partial }_x{u}_i^k{|}^{m-1}{\partial }_x{u}_i^k=u_{3-i}^{q_i},x=0,t>0;\hfill \\ \hfill & u_i(x,0)=u_{i0}\left(x\right),x\in {\mathbb{R}}_{+},\hfill \end{array}\right.\end{array}$$

(7)

where $m>1,k\ge 1,q_i,p_i>0,$ and $i=1,2,$ are numerical parameters.

They studied the global solvability, unsolvability of (7), and they found the following critical curves, which guaranteed the existence of a global solution:

(i)

${\left({\prod }_{i=1}^2{q}_i\right)}_c\le {\prod }_{i=1}^2{\displaystyle \frac{m(k+1-p_i)}{m+1}};$

(ii)

$min\left\{q_i(m+1)-{\displaystyle \frac{m(p_{3-i}-1)(p_i+k)}{p_i-1}}\right\}=0.$

Furthermore, they also show blow-up solutions for some large enough initial data.

The authors of [37] obtained similar results for the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\partial }_t{u}_i=\nabla \left(\right|\nabla u_i^{m_i}{|}^{p_i-2}\nabla u_i^{m_i}),x\in {\mathbb{R}}_{+}^N,t>0;\hfill \\ \hfill & -|\nabla u_i^{m_i}{|}^{p_i-2}\nabla u_i^{m_i}=u_1^{{\beta }_i}{u}_2^{q_i},x=0,t>0;\hfill \\ \hfill & u_i(x,0)=u_{i0}\left(x\right),x\in {\mathbb{R}}_{+}^N,\hfill \end{array}\right.\end{array}$$

where $m_i>1,p_i>1+1/m_i,{\beta }_i,q_i>0,i=1,2.$

They showed critical curves as follows:

(i)

$0<q_1{\beta }_2\le {\prod }_{i=1}^k\left[(m_i+1)(1-1/p_i)-{\beta }_i(2-i)-q_{3-i}(i-1)\right];$

(ii)

$min\{N{\lambda }_i-{\alpha }_i\}=0,i=1,2.$

Z.Li and C.Mu also considered these coupled systems through boundary flux in a more general case:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\partial }_t{u}_i={\partial }_x\left(\right|{\partial }_x{u}_i{|}^{m_i-1}{\partial }_x{u}_i),(x,t)\in {\mathbb{R}}_{+}\times (0,+\infty );\hfill \\ \hfill & -|{\partial }_x{u}_i{|}^{m_i-1}{\partial }_x{u}_i=u_{i+1}^{p_i},u_{k+1}≔u_1,x=0,t\in (0,+\infty );\hfill \\ \hfill & u_i(x,0)=u_{i0}\left(x\right),x\in {\mathbb{R}}_{+},\hfill \end{array}\right.\end{array}$$

where $0<m_i<1,p_i>0,i=1,2,\dots ,k,k\in \mathbb{N}.$

They stated the critical curves as follows:

(i)

${\left({\prod }_{i=1}^k{p}_i\right)}_c={\prod }_{i=1}^k{\displaystyle \frac{2m_i}{m_i+1}};$

(ii)

$min\{{\beta }_i-{\alpha }_i\}=0,$

where ${\beta }_i={\displaystyle \frac{(1-m_i){\alpha }_i+1}{1+m_i}},{\alpha }_{i+1}={\displaystyle \frac{(2{\alpha }_i+1)m_i}{(1+m_i)p_i}},{\alpha }_{k+1}={\alpha }_1,i=1,2,\dots ,k.$

Furthermore, the fundamental solution [38] to the single-equation case of system (1) with the Cauchy problem has been intensively studied by M. Bobokandov et al.

Although diffusion systems in homogeneous media and with density in the form $\rho \left(x\right)={\left|x\right|}^n$ have been widely studied, the case of an inhomogeneous density in the form ${(1+|x\left|\right)}^n$ coupled with nonlinear boundary interactions in cross-diffusion systems has received comparatively little attention. The essential problems of global solutions and finite-time blow-up [39] drive our study. Specifically, our goal is to understand how the exponents p, $m_i$, $q_i$, and n interact with the outcomes of the solution. We seek critical exponents to distinguish between global solvability and the blow-up case for the cross-diffusion system [24].

A fundamental analytical challenge in cross-diffusion systems is the loss of uniform ellipticity caused by nonlinear coupling. In the present model, the diffusion tensor degenerates both when spatial gradients vanish and when one of the solution components approaches zero. Consequently, the operator fails to be coercive in the classical $W^{1,p}$ sense, and standard compactness arguments based on uniformly elliptic diffusion are no longer applicable. Nevertheless, weak solutions can be formulated within degenerate Sobolev spaces compatible with the p-Laplacian structure. Similar functional settings and coercivity limitations have been extensively studied in the context of cross-diffusion systems by A.Jüngel [40] and A.Erhardt [11]. In these works, weak compactness is recovered not through uniform ellipticity but via a combination of monotonicity properties, truncation techniques, and a priori estimates. More precisely, local spatial bounds in $W_{\mathrm{loc}}^{1,p}\left({\mathbb{R}}_{+}\right)$, together with the control of time derivatives in negative Sobolev spaces, allow the application of compactness results of Aubin–Lions type adapted to degenerate or weighted frameworks. As a result, one can extract strongly convergent subsequences in $L_{\mathrm{loc}}^1({\mathbb{R}}_{+}\times (0,T))$, despite the absence of full coercivity.

An additional compactness mechanism arises from the finite propagation property inherent to degenerate diffusion. The non-negativity and boundedness of solutions imply that compactly supported initial data remain locally and compactly supported for finite times. This localisation effect prevents the loss of compactness at spatial infinity and plays a crucial role in the analysis of degenerate parabolic equations [17,28]. Taken together, these mechanisms—weighted energy estimates, monotonicity of diffusion operators, weak time regularity, and finite propagation—ensure that the coupling between diffusions preserves the weak compactness of the solution set, and they provide an analytical basis for limiting in weak formulations, even in the presence of strong degeneracy and nonlinear boundary interactions [11,40,41].

Throughout this paper, solutions to the cross-diffusion system are, therefore, understood in the weak sense. Specifically, for a given $T>0$, a pair $(u,v)$ is said to be a weak solution on ${\mathbb{R}}_{+}\times (0,T)$ if

$$0\le u,v\in L^{\infty }\left(0,T;L_{\mathrm{loc}}^1\left({\mathbb{R}}_{+}\right)\right)\cap L^p\left(0,T;W_{\mathrm{loc}}^{1,p}\left({\mathbb{R}}_{+}\right)\right),$$

with spatial derivatives

$${\partial }_{x}u,{\partial }_{x}v\in L_{\mathrm{loc}}^p({\mathbb{R}}_{+}\times (0,T)),$$

and time derivatives satisfying

$$\rho \left(x\right){\partial }_{t}u,\rho \left(x\right){\partial }_{t}v\in L_{\mathrm{loc}}^1\left(0,T;W_{\mathrm{loc}}^{-1,p^{\prime }}\left({\mathbb{R}}_{+}\right)\right),p^{\prime }={\displaystyle \frac{p}{p-1}}.$$

The nonlinear Neumann boundary conditions are satisfied in the sense of traces, which are well-defined under the above regularity assumptions. Due to the degeneracy of the diffusion operator and the cross-diffusion coupling, classical solutions are not expected in general. However, the weak formulation is well-posed in the stated function spaces and is fully compatible with the construction of self-similar sub- and super-solutions employed in the analysis [42,43].

2. Materials and Methods

The results shown in this paper are based on models developed from purely theoretical and analytical mathematical methods. The primary techniques employed include a self-similar analysis of partial differential equations and nonlinear separation of variables. These methods are applied to reduce the cross-diffusion system (1)–(3) into a set of coupled ordinary differential equations. The comparison principle is applicable at the level of weak solutions, provided the diffusion operators satisfy a p-Laplacian monotonicity structure and the cross-diffusion coefficients are non-negative [17,24,44]. This applicability relies on the fact that such a structure ensures the operator is strictly monotone in the sense of Minty–Browder [45]. This monotonicity is the fundamental requirement, as it guarantees that the operator generates a coercive mapping that preserves the order of solutions in the dual space. It is precisely this algebraic stability that allows us to successfully recover comparison arguments for degenerate parabolic equations, even where classical pointwise validity fails [11,24]. Moreover, the construction of explicit super-solutions and sub-solutions establishes the proofs of the main theorems. The comparison principle is applied to establish the conditions for global solvability versus finite-time blow-up.

3. Results

3.1. Global Solutions

Theorem 1. 

Let, $q_i\le 1,(i=1,2)$, then every nontrivial solution of the problems (1)–(3) is global in time.

Proof. 

To prove Theorem 1 and show an estimate of the solution or super-solution of the problem (1)–(3), we seek the solution as follows:

$$\begin{array}{c}\hfill {\widehat{u}}_i(x,t)=e^{L_{i}t}\left(K+e^{-M_i{\xi }_i}\right),{\xi }_i=(1+|x\left|\right)e^{-J_{i}t},\end{array}$$

(8)

where $K>max\left\{{\parallel u_0\parallel }_{\infty },{\parallel v_0\parallel }_{\infty }\right\},$

$$\begin{array}{cc}\hfill & M_i={(K+1)}^{{\displaystyle \frac{q_i+1-m_i}{p-1}}},J_i={\displaystyle \frac{L_i(p-2)+L_{3-i}(m_i-1)}{p+1+n}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & L_i=M_i^{p-1}{(K+1)}^{m_i-1}((p-1)+\left(m_i-1\right){\left(K+1\right)}^{{\displaystyle \frac{m_i-m_{3-i}+q_{3-i}-q_i}{m_{3-i}-1-q_{3-i}}}}),i=1,2.\hfill \end{array}$$

Using the approach employed in [46,47], we show that (8) is the super-solution to the problem (1)–(3). According to this approach, and comparing principles [25], the super-solution must fulfil the following systems:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \rho \left(x\right){\partial }_t{\widehat{u}}_i\ge {\partial }_x\left({\widehat{u}}_{3-i}^{m_i-1}{\left|{\partial }_x{\widehat{u}}_i\right|}^{p-2}{\partial }_x{\widehat{u}}_i\right),x\in {\mathbb{R}}_{+},t>0,\hfill \\ \hfill & -{\widehat{u}}_{3-i}^{m_i-1}{\left|{\partial }_x{\widehat{u}}_i\right|}^{p-2}{\partial }_x{\widehat{u}}_i{|}_{x=0}\le {\widehat{u}}_i^{q_i}(0,t),t>0,\hfill \\ \hfill & {\widehat{u}}_1(x,0)\ge u_0\left(x\right),x\in {\mathbb{R}}_{+},\hfill \\ \hfill & {\widehat{u}}_2(x,0)\ge v_0\left(x\right),x\in {\mathbb{R}}_{+}.\hfill \end{array}\right.\end{array}\end{array}$$

(9)

After substituting (8) into (1)–(3), the following inequalities yield

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & -{\widehat{u}}_{3-i}^{m_i-1}|{\partial }_x{\widehat{u}}_i{|}^{p-2}{\partial }_x{\widehat{u}}_i{|}_{x=0}=M_i^{p-1}{e}^{((L_i-J_i)(p-1)+L_{i-3}(m_i-1))t-m_i(p-1){\xi }_i}·\hfill \\ \hfill & {(K+e^{-M_{3-i}{\xi }_{3-i}})}^{m_i-1}{|}_{{\xi }_i=e^{-J_{i}t}}\le e^{q_i{L}_{i}t}{\left(K+e^{-M_i{\xi }_i}\right)}^{q_i}{|}_{{\xi }_i=e^{-J_{i}t}},\hfill \end{array}\end{array}$$

(10)

hence, we choose $M_i$ such that

$$\begin{array}{c}\hfill M_i^{p-1}{(K+1)}^{m_i-1}={(K+1)}^{q_i},(L_i-J_i)(p-1)+L_{i-3}(m_i-1)\le q_i{L}_i,\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\partial }_x\left({\widehat{u}}_{3-i}^{m_i-1}{\left|{\partial }_x{\widehat{u}}_i\right|}^{p-2}{\partial }_x{\widehat{u}}_i\right)=M_i^{p-1}{e}^{((L_i-J_i)(p-1)+L_{i-3}(M_i-1))t-M_i(p-1){\xi }_i}(K+\hfill \\ \hfill & e^{-M_{3-i}{\xi }_{3-i}}{)}^{m_i-2}\left[M_i(p-1)e^{-J_{i}t}(K+e^{-M_{3-i}{\xi }_{3-i}})+M_{3-i}(M_i-1)e^{-J_{3-i}t}\right]\le \hfill \\ \hfill & M_i^{p-1}{e}^{(L_i(p-1)-J_{i}p+L_{i-3}(m_i-1))t}{(K+1)}^{m_i-2}[M_i(p-1)(K+1)+M_{3-i}(M_i-1)];\hfill \\ \hfill & \rho \left(x\right){\partial }_t{\widehat{u}}_i=L_i{\xi }_i^n{e}^{(L_i+J_i(n+1))t}(K+e^{-M_i{\xi }_i})+M_i{J}_i{\xi }_i^{n+1}{e}^{(L_i+J_n)t-M_i{\xi }_i}\ge \hfill \\ \hfill & L_i{\xi }_i^n{e}^{(L_i+J_i(n+1))t}(K+1),\hfill \end{array}\end{array}$$

and to fulfil these inequalities, we choose $J_i$ and $L_i$ such that

$$\begin{array}{cc}\hfill & L_i(p-1)-J_{i}p+L_{i-3}(m_i-1)=L_i+J_i(n+1),\hfill \end{array}$$

$$\begin{array}{cc}& M_i^{p-1}{(K+1)}^{m_i-2}[M_i(p-1)(K+1)+M_{3-i}(M_i-1)]=L_i,\hfill \end{array}$$

and for initial conditions, we obtain the following inequalities:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\widehat{u}}_i(x,0)=K+e^{-M_i(1+|x\left|\right)}>K\ge max\{u_0\left(x\right),v_0\left(x\right)\},\forall x\in {\mathbb{R}}_{+},\hfill \\ \hfill & K>max\{\parallel u_0\parallel ,\parallel v_0\parallel \}.\hfill \end{array}\end{array}$$

(11)

According to definition (8) and expressions (10)–(11), inequality (9) holds. The proof of Theorem 1 is completed. □

Theorem 2. 

Let $q_i>m_{3-i}+p-1,$ and $min\left\{\alpha (n+1)q_i\right\}\ge 1,(i=1,2),$ then every nontrivial solution of the problem (1)–(3) is global in time for sufficiently small initial data,

• where $\alpha ={\displaystyle \frac{(q_1-1)(q_2-1)-(q_1-1)(m_1-1)-(q_2-1)(p-2)}{(q_1-1)(q_2-1)(p+n)-(1+n)\left((q_1-1)(m_1-1)-(q_2-1)(p-2)\right)}}.$

Proof. 

We seek the $u,v$ in the self-similar form solution [25,48] to demonstrate Theorem 2 and show an upper estimate of the solution to the problem (1)–(3):

$$\begin{array}{c}\hfill {\underline{u}}_i(x,t)={(T+t)}^{-{\alpha }_i}{f}_i\left(\xi \right),\xi =(1+|x\left|\right){(T+t)}^{-\alpha },\end{array}$$

(12)

where $T>0,{\alpha }_i={\displaystyle \frac{1-\alpha (1+n)}{q_i-1}}.$

We put ${\underline{u}}_i(x,t)$ into (1)–(3) and obtain the following expression:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\xi }}\left(f_{3-i}^{m_i-1}|{\displaystyle \frac{d{f}_i}{d\xi }}{|}^{p-2}{\displaystyle \frac{d{f}_i}{d\xi }}\right)+\alpha {\xi }^{n+1}{\displaystyle \frac{d{f}_i}{d\xi }}+{\alpha }_i{\xi }^n{f}_i=0,\end{array}$$

(13)

and the cross-diffusion condition [19,49] for the system (1):

$$\begin{array}{c}\hfill (q_1-1)(m_1-p+1)=(q_2-1)(m_2-p+1).\end{array}$$

We seek the solution for system (13) in the Barenblatt profile form [50]

$$\begin{array}{c}\hfill {\overline{f}}_i\left(\xi \right)=A_i{\left(a-{\xi }^{\gamma }\right)}_{+}^{{\gamma }_i},\end{array}$$

where $\gamma ={\displaystyle \frac{p+n}{p-1}},{\gamma }_i={\displaystyle \frac{(p-1)(p-1-m_i)}{{\delta }_{\gamma }}},{\delta }_{\gamma }={(p-2)}^2-(m_1-1)(m_2-1)\ne 0,$  ${\left(d\right)}_{+}=max\{d,0\},a=const.\ge 0,A_i=\left|{\left(\alpha {\gamma }^{1-p}\right)}^{p-m_i-1}\right|{\displaystyle \frac{{\gamma }_{3-i}^{m_i-1}}{{\gamma }_i^{p-2}}}{|}^{p-1}{|}^{{\displaystyle \frac{1}{{\delta }_{\gamma }}}}.$

It is easy to see that $\overline{f}\left(\xi \right)$ satisfies the following system:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\xi }}\left({\overline{f}}_{3-i}^{m_i-1}|{\displaystyle \frac{d{\overline{f}}_i}{d\xi }}{|}^{p-2}{\displaystyle \frac{d{\overline{f}}_i}{d\xi }}+\alpha {\xi }^{n+1}{\overline{f}}_i\right)=0,\end{array}$$

which yields the following residue after substituting $\overline{f}\left({\xi }_i\right)$ into (13):

$$\begin{array}{c}\hfill {\xi }^n{\overline{f}}_i\left(\xi \right)\left[{\alpha }_i-\alpha (n+1)\right]\le 0.\end{array}$$

(14)

Based on the non-negative properties of ${\overline{f}}_i$ and $\xi$, the following inequality must be hold in order to fulfil (14):

$$\begin{array}{c}\hfill 0\le \alpha (n+1)-{\alpha }_i={\displaystyle \frac{\alpha (1+n)q_i-1}{q_i-1}}\Rightarrow \alpha (n+1)q_i\ge 1.\end{array}$$

The proof of Theorem 2 is completed. □

Corollary 1. 

If $\alpha (n+1)={\alpha }_i,$ then the self-similar solution (12) becomes the source-type exact solution to the system (1).

3.2. Local or Blow-Up Solutions

Theorem 3. 

Let $1<{\displaystyle \frac{1}{\lambda (n+1)}}\le min\left\{q_i\right\},(i=1,2),$ then every solution of the problem (1)–(3) blows up in a finite time for sufficiently large initial data.

• where $\lambda ={\displaystyle \frac{(q_i-1)(m_i-q_{3-i})+(p-2)(q_{3-i}-1)}{(n+1)[(p-2)(q_{3-i}-1)+(m_i-q_{3-i})(q_i-1)]-(p-1)(q_i-1)(q_{3-i}-1)}}>0.$

Proof. 

We look for the solution of problem (1)–(3) as follows:

$$\begin{array}{c}\hfill {\overline{u}}_i(x,t)={(T-t)}^{-{\lambda }_i}{g}_i\left(\xi \right),\xi =(1+|x\left|\right){(T-t)}^{-\lambda },\end{array}$$

(15)

where ${\lambda }_i={\displaystyle \frac{1-\lambda (n+1)}{q_i-1}}>0,i=1,2.$

After substituting (15) into (1)–(3), we obtain the following systems:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\xi }}\left(g_{3-i}^{m_i-1}|{\displaystyle \frac{d{g}_i}{d\xi }}{|}^{p-2}{\displaystyle \frac{d{g}_i}{d\xi }}\right)-\lambda {\xi }^{n+1}{\displaystyle \frac{d{g}_i}{d\xi }}-{\lambda }_i{\xi }^n{g}_i=0,\hfill \\ \hfill & -g_{3-i}^{m_i-1}|{\displaystyle \frac{d{g}_i}{d\xi }}{|}^{p-2}{\displaystyle \frac{d{g}_i}{d\xi }}{|}_{\xi ={(T-t)}^{\lambda }}=g_i^{q_i}\left(\xi \right){|}_{\xi ={(T-t)}^{\lambda }}.\hfill \end{array}\right.\end{array}$$

(16)

We look for the solution of system (16) as follows:

$$\begin{array}{c}\hfill \overline{g}\left(\xi \right)=B_i{(b+{\xi }^{\gamma })}^{{\gamma }_i},\end{array}$$

(17)

where $b>0,B_i=|{\displaystyle \frac{\lambda }{\alpha }}{|}^{{\displaystyle \frac{p-m_i-1}{{\delta }_{\gamma }}}}{A}_i.$

It is clear that function ${\overline{g}}_i\left(\xi \right)$ satisfies the following system:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{d\xi }}\left({\overline{g}}_{3-i}^{m_i-1}|{\displaystyle \frac{d{\overline{g}}_i}{d\xi }}{|}^{p-2}{\displaystyle \frac{d{\overline{g}}_i}{d\xi }}-\lambda {\xi }^{n+1}{\overline{g}}_i\right)=0,\end{array}$$

and after substituting that into (16), the following system yields

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & (\lambda (n+1)-{\lambda }_i){\xi }^n{\overline{g}}_i\ge 0,\hfill \\ \hfill & -\lambda {\xi }^{n+1}{\overline{g}}_i{|}_{\xi ={(T-t)}^{\lambda }}\le {\overline{g}}_i^{q_i}\left(\xi \right){|}_{\xi ={(T-t)}^{\lambda }},\hfill \end{array}\right.\end{array}$$

or

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \lambda (n+1)-{\lambda }_i={\displaystyle \frac{\lambda (n+1)q_i-1}{q_i-1}}\ge 0,\hfill \\ \hfill & -\lambda {\xi }^{n+1}{\overline{g}}_i{|}_{\xi ={(T-t)}^{\lambda }}\le 0\le {\overline{g}}_i^{q_i}\left(\xi \right){|}_{\xi ={(T-t)}^{\lambda }}.\hfill \end{array}\right.\end{array}$$

Also, we have ${\lambda }_i>0$ conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\displaystyle \frac{\lambda (n+1)q_i-1}{q_i-1}}\ge 0,\hfill \\ \hfill & {\displaystyle \frac{1-\lambda (n+1)}{q_i-1}}>0,\hfill \end{array}\right.\end{array}$$

(18)

Obviously, if the inequality $1<{\displaystyle \frac{1}{\lambda (n+1)}}\le min\left\{q_i\right\},(i=1,2),$ holds, then the system of inequalities (18) is also satisfied.

By virtue of the principle of comparison of solutions [25]—the constructed sub-solutions in (15)–(17) give an estimate for the initial data:

$$\begin{array}{c}\hfill {\overline{u}}_i(x,0)\ge B_i{T}^{-{\lambda }_i}{(b+(1+\left|x\right|)}^{\gamma }{T}^{-\lambda \gamma }{)}^{{\gamma }_i}.\end{array}$$

Therefore, the solution to problem (1)–(3) is unbounded from above:

$$\begin{array}{c}\hfill u_i(x,t)\ge {(T-t)}^{-{\lambda }_i}\overline{g}\left(\xi \right){|}_{\xi ={(T-t)}^{\lambda }}.\end{array}$$

Hence, we can see that the blow-up rate is equal to $-{\lambda }_i$. For numerical solutions and graphical representations of the blow-up rate, see [51,52], and for more information on blow-up solutions, see [53,54,55,56,57] and the references therein. Moreover, we conclude that Theorem 3 holds for systems (1)–(3). □

4. Discussion

In this paper, we establish a clear qualitative classification of solution behaviour for the considered cross-diffusion system with nonlinear boundary flux. Specifically, when $q_i\le 1$$(i=1,2)$, every nontrivial solution exists globally in time for arbitrary admissible initial data. In contrast, for $q_i>1$, the dynamics exhibit a threshold phenomenon: solutions corresponding to sufficiently small initial data remain global, whereas large initial data lead to finite-time blow-up. This dichotomy highlights the fundamental role of the boundary flux exponents in determining the long-time behaviour of solutions and confirms the presence of Fujita-type critical mechanisms in the present double nonlinearities and the density variable function. Despite these results, several important questions remain open. In view of the finite propagation property induced by degenerate diffusion, it is plausible that the blow-up occurs either at the boundary $x=0$, where the nonlinear flux is imposed, or at interior points associated with maxima of the initial data. Determining whether blow-up is localised or boundary-driven remains an open question and is closely related to understanding the interaction between cross-diffusion and boundary nonlinearities.

Further directions include the study of uniqueness and the stability of weak solutions in the presence of cross-diffusion coupling as well as the extension of the present results to higher-dimensional domains. In such settings, the interplay between degeneracy and boundary effects may give rise to new critical phenomena and pattern-forming mechanisms. Finally, numerical investigations could provide valuable insight into the sharpness of theoretical conditions and help visualise blow-up profiles and interfaces, thereby complementing the analytical results.

5. Conclusions

In this paper, we investigate the qualitative properties of solutions for a double nonlinear, a cross-diffusion system in a non-homogeneous medium, defined by Equations (1)–(3). In contrast to previous works, this paper considers a diffusion system with a spatially dependent density given by $\rho \left(x\right)={(1+|x\left|\right)}^n$ and includes nonlinear boundary fluxes. Our analysis, based on the construction of self-similar sup- and sub-solutions, establishes a complete classification of solution behaviour based on the boundary flux exponents $q_i,(i=1,2).$ This analysis validates the self-similar method for this class of complex problems.

The research effectively illustrates the usefulness of self-similar analysis and the Barenblatt profile as a potent analytical instrument. The effective application of the etalon equations to systems (1)–(3) with this combined complexity, such as cross-diffusion, non-homogeneity, double nonlinearity, etc., validates the technique for a broader class of degenerate problems.