On Classical Solutions for A Kuramoto—Sinelshchikov—Velarde-Type Equation

: The Kuramoto–Sinelshchikov–Velarde equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane ﬂame propagation, taking into account the combined inﬂuence of diffusion and thermal conduction of the gas on the stability of a plane ﬂame front. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem.

Taking κ = r = δ = µ = β = γ = 0 and using the variable (7), (1) becomes which is known as the modified Korteweg-de Vries equation. [10][11][12][13][14][15] show that (9) is a non-slowly varying envelope approximation model that describes the physics of few-cycle-pulse optical solitons. In [3,5], the Cauchy problem for (9) is studied, while in [9,16], the convergence of the solution of (9) to the unique entropy solution of the following scalar conservation law Assuming κ = 1 and q = r = γ = 0, (1) reads Equation (11) arises in interesting physical situations, for example as a model for long waves on a viscous fluid owing down an inclined plane [17] and to derive drift waves in a plasma [18]. Equation (11) was derived also independently by Kuramoto [19][20][21] as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky [22] as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front.
The dynamical properties and the existence of exact solutions for (11) have been investigated in [29][30][31][32][33][34]. In [35][36][37], the control problem for (11) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [38], the problem of global exponential stabilization of (11) with periodic boundary conditions is analyzed. In [39], it is proposed a generalization of optimal control theory for (11), while in [40] the problem of global boundary control of (11) is considered. In [41], the existence of solitonic solutions for (11) is proven. In [1,42], the well-posedness of the Cauchy problem for (11) is proven, using the energy space technique and the fixed-point method, respectively. In particular, in [1], the well-posedness is proven, under the assumption Observe that thanks to (7), Equation (11) is equivalent to the following one Consequently, following [8,9,43], in [44], it is proven that when δ, β 2 go to zero, the solution of (13) converges to the unique entropy one of the Burgers equation.
From a mathematical point of view, in [47] the exact solutions for the KV equation are studied, while in [48], the initial boundary problem is analyzed. In [49], the well-posedness of the Cauchy problem for the KV equation is proven in the energy spaces.
The main result of this paper is the following theorem.
Theorem 1. Let T > 0 be given. The following statements hold.
Theorem 1 improves the existing literature (see [1]) because it gives the well-posedness of (1) under Assumption (2), without additional assumption on the constants. Under Assumptions (3) and (4), Theorem 1 gives only the existence of the solution, while the uniqueness is guaranteed by Assumption (5). The argument of Theorem 1 relies on deriving suitable a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem [50]. We conjecture that our argument can be applied also to the the initial boundary value problem and to multidimensional version of the problem.

Proof of Theorem 1, under the Assumptions
Let us prove some a priori estimates on u. We denote with C 0 the constants which depend only on the initial data, and with C(T), the constants which depend also on T.
We prove the following lemma.
Lemma 1. Fix T > 0. Then, we have that for every 0 ≤ t ≤ T.
Proof. Let 0 ≤ t ≤ T. Multiplying (17) by −2∂ 2 x u, an integration on R gives . Therefore, we have that Observe that Consequently, by the Young inequality, .
It follows from (19) that The Gronwall Lemma and (4) give that is (18).
The proof of the previous lemma is based on the regularity of the functions u and the following result.
In particular, we have that for every 0 ≤ t ≤ T.
There exists a constant C(T) > 0, such that for every 0 ≤ t ≤ T.
Proof. Let 0 ≤ t ≤ T. Multiplying (17) by 2∂ t u, an integration on R gives Due to the Young inequality, where D 2 is a positive constant, which will be specified later. It follows from (35) that .

Proof of Theorem 1, under the assumptions
Let us prove some a priori estimates on u. We denote with C 0 the constants which depend only on the initial data, and with C(T), the constants which depend also on T. Lemma 6. Fix T > 0 and assume (4) or (5). There exists a constant C(T) > 0, such that for every 0 ≤ t ≤ T.
Proof. Let 0 ≤ t ≤ T. Multiplying (44) by 2u, an integration on R gives Due to the Young inequality, .
Proof. Let 0 ≤ t ≤ T. Multiplying (44) by 2∂ t u, an integration on R gives Due to (21) and the Young inequality, where D 3 is a positive constant, which will be specified later. Consequently, by (50), .

Lemma 9.
Fix T > 0 and assume (5). There exists a positive constant C(T) > 0, such that for every 0 ≤ t ≤ T. In particular, we have that Proof. Let 0 ≤ t ≤ T. Multiplying (44) by −2∂ 6 x u, an integration on R gives Therefore, we have that Due to (21), (28), (49) and the Young inequality, where D 4 is a positive constant, which will be specified later. Therefore, by (53), Observe that Thus, by the Young inequality, .
It follows from (54) that .

Conclusions
In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem of the Kuramoto-Sinelshchikov-Velarde equation, that describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking in account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. Our result requires very general assumptions on the coefficients and the argument is based on energy estimates and the Cauchy-Kovalevskaya Theorem.