H 1 Solutions for a Kuramoto–Velarde Type Equation

. Kuramoto–Velarde equation describes the spatiotemporal evolution of the morphology of steps on crystal surfaces, or the evolution of the spinoidal decomposition of phase separating systems in an exter-nal ﬁeld. We prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation for each choice of the terminal time T .


Introduction
In this paper, we investigate the well-posedness of the following Cauchy problem: with γ, α, κ, δ, β ∈ R, β = 0. (1.2) On the flux f ), we assume for some positive constant C 0 . On the initial datum, we assume u 0 ∈ H 1 (R), u 0 = 0. (1.4) Taking 6) It is known as the Kuramoto-Velarde equation and it describes slow spacetime variations of disturbances at interfaces, diffusion-reaction fronts and plasma instability fronts [26,30,31]. It also describes Benard-Marangoni cells that occur when there is large surface tension on the interface [38,67,71] in a microgravity environment. This situation arises in crystal growth experiments aboard an orbiting space station, although the free interface is metastable with respect to small perturbations. The nonlinearities κu∂ 2 x u and δ(∂ x u) 2 model pressure destabilization effects striving to rupture the interface. (1.6) is deduced in [66] to describe the long waves on a viscous fluid owing down an inclined plane, and in [25] to model the drift waves in a plasma.
From a mathematical point of view, in [40], the exact solutions for (1.6) are studied, while, in [60], the initial boundary problem is analyzed. In [7,8], the authors prove the existence of the solitons for (1.6). Instead, in [56], the existence of traveling wave solutions for (1.6) is studied. In [41], the author analyzes the existence of the periodic solution for (1.6), under appropriate assumptions on b 1 , γ, α, κ, δ and β. The well-posedness of the Cauchy problem for (1.6) is proven in [59], using the energy space technique and taking b 1 = 0. In [15], under assumption u 0 ∈ H 2 (R), u 0 = 0, (1.7) and δ = 2κ, through a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem, the well-posedness of the classical solutions of (1.6) is proven. In [11], under Assumption (1.7) and δ = 2κ, the wellposedness of classical solutions is proven, under appropriate assumptions on β, T and H 1 -norm of the initial datum. Finally, in [16], the well-posedness of classical solutions is proven, under Assumption (1.7) and under appropriate assumptions on β, T and L 2 -norm of the initial datum. Taking κ = δ = 0 in (1.6), we have It was also independently deduced by Kuramoto [45][46][47] to describe the phase turbulence in reaction-diffusion systems, and by Sivashinsky [64], to describe 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. Equation (1.8) can be used to study incipient instabilities in several physical and chemical systems [5,37,48]. Moreover, (1.8), which is also known as the Benney-Lin equation [3,52], was derived by Kuramoto in the study of phase turbulence in Belousov-Zhabotinsky reactions [51].
From a mathematical point of view, the dynamical properties and the existence of exact solutions for (1.8) have been investigated in [29,42,44,57,58,69]. The control problem for (1.8) are studied in [1,4,32]. In [6], the problem of global exponential stabilization of (1.8) with periodic boundary conditions is analyzed. In [39], it is proposed a generalization of optimal control theory for (1.8), while, in [55], the problem of global boundary control of (1.8) is considered. In [61], the existence of solitonic solutions for (1.8) is proven. In [2,15,17,18,65], the well-posedness of the Cauchy problem for (1.8) is proven, using the energy space technique, the fixed point method, a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem and a priori estimates together with an application of the Aubin-Lions Lemma, respectively. Instead, in [19,53,54], the initial-boundary value problem for (1.8) is studied, using a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem, and the energy space technique, respectively. Finally, following [20,49,62], in [21], the convergence of the solution of (1.8) to to the unique entropy one of the Burgers equation is proven. Taking It models the spinodal decomposition of phase separating systems in an external field [28,50,70], the spatiotemporal evolution of the morphology of steps on crystal surfaces [33,43,61] and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension [34][35][36]. In the case of a growing crystal surface with strongly anisotropic surface tension, the function u represents the surface slope, while the constants b 1 and b 1 are the growth driving forces proportional to the difference between the bulk chemical potentials of the solid and fluid phases. Equation (1.10) is also deduced by Watson [68] as a small-slope approximation of the crystal growth model obtained in [27].
Here we complete the results of [11,16]. Here we assume (1.4) on the initial condition and our arguments are based on the Aubin-Lions Lemma [13,14,23,24,63].
The paper is organized as follows. In Sect. 2, we prove several a priori estimates on a vanishing viscosity approximation of (1.1). Those play a key role in the proof of our main result, that is given in Sect. 3.

Vanishing Viscosity Approximation
Our existence argument is based on passing to the limit in a vanishing viscosity approximation of (1.1).
Fix a small number 0 < ε < 1 and let u ε = u ε (t, x) be the unique classical solution of the following problem [9][10][11]: x ∈ R, where u ε,0 is a C ∞ approximation of u 0 , such that where C 0 is a positive constant, independent on ε.
Let us prove some a priori estimates on u ε . We denote by C 0 the constants which depend only on the initial data, and by C(T ), the constants which depend also on T .
Proof. We begin by proving that e (40AC 0 +a 2 1 )t 2β 2 for every 0 ≤ t ≤ T , where A is a arbitrary positive constant, and It follows from (2.10), (2.11) and (2.12) that Due to (1.3) and the Young inequality, where D 1 is a positive constant, which will be specified later. Therefore, by . (2.14) Observe that Thanks to the Young inequality, It follows from (2.14) that We define (2.16) It follows from (2.15) that Thanks to the Hölder inequality, Hence, Consequently, by (2.19), (2.20) it follows from (2.20) that that is, (2.23) Multiplying (2.23) by e 2 4t , we get Therefore, d dt Integrating on (0, t), we have that that is, Consequently, we have that which gives, 1
Finally, we prove (2.35). We begin by observing that (2.36) Due to the (2.4) and the Hölder and Young inequalities, Due to the Young inequality, Consequently, by (2.37), we have that Thanks to the Hölder inequality, Hence, Observe that Thanks to the Hölder inequality, It follows from (2.39) that (2.41) Consequently, by (2.38) and (2.41), we get Integrating on (0, t), we have (2.35).

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1.
Funding Open access funding provided by Politecnico di Bari within the CRUI-CARE Agreement.

Data Availability Statement
The manuscript has no associate data.

Conflict of Interest
The authors declare that they do not have any conflict of interest.