Well-posedness of the classical solution for the Kuramto–Sivashinsky equation with anisotropy effects

The Kuramto–Sivashinsky equation with anisotropy effects models the spinodal decomposition of phase separating systems in an external field, the spatiotemporal evolution of the morphology of steps on crystal surfaces and the growth of thermodynamically unstable crystal surfaces with strongly anisotropic surface tension. Written in terms of the step slope, it can be represented in a form similar to a convective Cahn–Hilliard equation. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

(1.3) Observe that, using the variable (see [24,57]) v = ∂ x u, (1.4) Equation (1.1) is equivalent to the following one: which is known as the convective Cahn-Hilliard equation (see [24,33]). From a physical point of view, (1.1) and (1.5) model the spinodal decomposition of phase separating systems in an external field [19,42,64], the spatiotemporal evolution of the morphology of steps on known as the Cahn-Hilliard equation [8,9,50,51]. It describes the process of spinodal decomposition. In this case, the function u is the concentration of one of the components of an alloy. [51] shows that (1.6) has an exact solution that describes the final stage of the spinodal decomposition, the formation of the interface between two stable state of an alloy with different concentrations. It also describes the coarsening dynamics of the faceting of thermodynamically unstable surfaces [31,56]. Moreover, [34] shows that Eq. (1.6) can be an effective tool in technological applications to design nanostructured materials.
From a mathematical point of view, in [2], the existence of some extremely slowly evolving solutions for (1.5) is proven, considering a bounded domain, while, in [6,22], the problem of a global attractor is studied. Instead, in [27,65], numerical schemes for (1.5) are analyzed, while, in [60], an approximate analytical solution is studied.
Observe that Eq. (1.5) is has been studied in the multidimensional case in the papers [7,18,66] and their references.
Taking τ = κ = δ = 0 in (1.5), we have the following equation (1.7) describes a spinodal decomposition in the presence of an external (e.g., gravitational or electric) field, when the dependence of the mobility factor on the order parameter is important [19,24,42,64]. From a mathematical point of view, the coarsening dynamics for (1.7) has been studied in the limit 0 < q 1 in [19,26] and analytically in [62]. In [1], a numerical scheme is studied for (1.7), while the existence of the periodic solution are analyzed in [20,36]. In [42,47], the existence of exact solutions for (1.7) and its viscous form have been investigated. Moreover, [26] shows that, when q → ∞, (1.7) reduces to the Kuramoto-Sivashinsky equation (see Eq. (1.8)). Physically, it means that, with the growth of the driving force, there must be a transition from the coarsening dynamics to a chaotic spatiotemporal behavior.
Assuming γ = τ = κ = δ = 0 in (1.5), we have the following equation: (1.8) arises in interesting physical situations, for example as a model for long waves on a viscous fluid flowing down an inclined plane [59] and to derive drift waves in a plasma [16]. Equation (1.8) was derived also independently by Kuramoto [37][38][39] as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky [55] 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. Equation (1.8) also describes incipient instabilities in a variety of physical and chemical systems [11,29,40]. Moreover, (1.8), which is also known as the Benney-Lin equation [4,43], was derived by Kuramoto in the study of phase turbulence in the Belousov-Zhabotinsky reaction [44].
The dynamical properties and the existence of exact solutions for (1.8) have been investigated in [21,32,35,48,49,63]. In [3,10,23], the control problem for (1.8) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [12], the problem of global exponential stabilization of (1.8) with periodic boundary conditions is analyzed. A generalization of optimal control theory for (1.8) was proposed in [30], while in [45] the problem of global boundary control of (1.8) is considered. In [53], the existence of solitonic solutions for (1.8) is proven. In [5,13,57], the well-posedness of the Cauchy problem for (1.8) is proven, using the energy space technique, a priori estimates together with an application of the Cauchy-Kovalevskaya and the fixed point methods, respectively. Finally, following [14,41,54], in [15], the convergence of the solution of (1.8) to the unique entropy one of the Burgers equation is proven when α, β → 0.
Before stating our main result it is important to comment our assumption (1.2) on the coefficients. That condition guarantees the conservation of the H 2 norm of the solution in time, in other words thanks to (1.2) the map t → u(t, ·) never leaves the energy space, that is H 2 .
We use the following definition of solution.
and for every test function with compact support The main result of this paper is the following theorem. (1.10) Moreover, if u 1 and u 2 are two solutions of (1.1), we have that for some suitable C(T ) > 0, and every 0 ≤ t ≤ T . Assuming (1.2) and u 0 ∈ H 3 (R), δ = 0, (1.12) there exists a unique solution u of (1.1), such that (1.13) Moreover, if u 1 and u 2 are two solutions of (1.1), we have that

A priori estimates
In this section, we 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 begin by proving the following result Therefore, we have that (2.5) Due to the Young inequality, where D 1 is a positive constant, which will be specified later. It follows from (2.5) that We search D 1 such that, By (2.7), we have that Thanks to (1.2), D 1 does exist. Therefore, by (1.2), (2.6), (2.7) and (2.8), we have that where K 2 1 , K 2 2 are two appropriate positive constants. Due to the Young inequality, (2.10) Observe that Therefore, by the Young inequality, (2.11) Consequently, by (2.10), (2.12) Integrating on (0, t), by the Gronwall Lemma and (1.3), we have that Integrating on (0, t), by (2.2), we have (2.4).
The proof of this lemma is based on the following result. Proof. We begin by observing that

Lemma 2.3. We have that
By the Young inequality, It follows from (2.18) that which gives (2.17).
Therefore, we have that d dt Due to the Young inequality, where D 3 is a positive constant, which will be specified later. It follows from (2.27) that Taking Due to the Young inequality, where D 4 is a positive constant, which will be specified later. Therefore, by (2.28), (2.29) We prove (2.25). Thanks to (2.2), (2.29) and the Hölder inequality, Therefore, Taking we have that Therefore, we have that d dt Due to (2.2), (2.25), (2.26) and the Young inequality, where D 5 is a positive constant, which will be specified later. It follows from (2.32) that .

Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. We begin by proving the following lemma. We prove (1.11). Let u 1 and u 2 be two solutions of (1.1), which verify (1.10), that is Then, the function is the solution of the following Cauchy problem: x ∈ R. (3.2) Observe that (3.2) is equivalent the following one: Observe that, since u 1 , u 2 ∈ H 4 (R), for every 0 ≤ t ≤ T , we have that Thanks to (3.4), we obtain Due to (3.4), (3.5) and the Young inequality, (3.7) Observe that Therefore, by the Young inequality, Proof. We begin by observing that, since δ = 0, (1.1) reads  We prove (1.14). Let u 1 and u 2 be two solutions of (3.9), which satisfy (1.13), that is
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