Well-posedness result for the Kuramoto–Velarde equation

The Kuramoto–Velarde equation describes slow space-time variations of disturbances at interfaces, diffusion–reaction fronts and plasma instability fronts. It also describes Benard–Marangoni cells that occur when there is large surface tension on the interface in a microgravity environment. Under appropriate assumption on the initial data, of the time T, and the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

On the initial datum, we assume u 0 ∈ H 2 (R), u 0 = 0 ( 1 . 2 ) and one of the following where . (1.7) From a physical point of view, Eq. (1.1), known as the Kuramoto-Velarde equation, describes slow space-time variations of disturbances at interfaces, diffusion-reaction fronts and plasma instability fronts [1][2][3]. It also describes Benard-Marangoni cells that occur when there is large surface tension on the interface [4][5][6] 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. In particular, the nonlinearities, γ (∂ x u) 2 and αu∂ 2 x u, model pressure destabilization effects striving to rupture the interface. Moreover, in [7], (1.1) is deduced to describe the long waves on a viscous fluid flowing down an inclined plane, while, in [8], (1.1) is deduced to model the drift waves in a plasma.
From a mathematical point of view, in [22], the exact solutions for (1.1) are studied, while in [23], the initial boundary problem is analyzed. In [1,24], the existence of the solitons is proven, while in [25], the existence of traveling wave solutions for (1.1) is analyzed. In [26], the author analyzes the existence of the periodic solution for (1.1), under appropriate assumptions on κ, ν, δ, β, γ , α. The well-posedness of the Cauchy problem for (1.1) is proven in [27], using the energy space technique and assuming κ = 0, and in [28], through a priori estimates together with an application of the Cauchy-Kovalevskaya and choosing γ = 2α. (1.8) In particular, in [27], the author gives some suitable conditions on ν, δ, β, γ , α, and prove the local well-posedness of (1.1), with κ = 0. Instead, in [28], under Assumptions (1.2) and (1.8), the authors prove well-posedness of (1.1), for each choose of β and T .
Observe that (1.1) generalizes the following equation: that (1.9) was also independently deduced by Kuramoto [29][30][31] to describe the phase turbulence in reaction-diffusion systems, and by Sivashinsky [32] 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 (1.9) can be used to study incipient instabilities in several physical and chemical systems [33][34][35]. Moreover, (1.9), which is also known as the Benney-Lin equation [36,37], was derived by Kuramoto in the study of phase turbulence in Belousov-Zhabotinsky reactions [38].
The dynamical properties and the existence of exact solutions for (1.9) have been investigated in [39][40][41][42][43][44]. In [45][46][47], the control problem for (1.9) with periodic boundary conditions, and on a bounded interval are studied, respectively. In [48], the problem of global exponential stabilization of (1.9) with periodic boundary conditions is analyzed. In [49], it is proposed a generalization of optimal control theory for (1.9), while in [50] the problem of global boundary control of (1.9) is considered. In [51], the existence of solitonic solutions for (1.9) is proven. In [28,[52][53][54], the well-posedness of the Cauchy problem for (1.9) 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 [55][56][57], the initial-boundary value problem for (1.1) is studied, using a priori estimates together with an application of the Cauchy-Kovalevskaya Theorem, and the energy space technique, respectively. Finally, following [58][59][60], in [61], the convergence of the solution of (1.9) to the unique entropy one of the Burgers equation is proven.
The main result of this paper is the following theorem.
Moreover, if u 1 and u 2 are two solutions of (1.1) in correspondence of the initial data u 1,0 and u 2,0 , we have that for some suitable C > 0, and every 0 ≤ t ≤ T .
Compared to [27], Theorem 1.1 gives some conditions on u 0 , β and T to have classical solutions for (1.1), under Assumption (1.2). Moreover, the argument of Theorem 1.1 relies on deriving suitable a priori estimates together with the existence result in [28]. The paper is organized as follows. In Sect. 2, we prove some a priori estimates of (1.1), under Assumptions (1.3), (1.4), (1.5) and (1.6), respectively. Those play a key role in the proof of our main result, which is given in Sect. 3.

A priori estimates
In this section, we prove some a priori estimates on u.
We prove the following result.

Lemma 2.1
We have that for every 0 ≤ t ≤ T . Moreover, Observe that Consequently, by (2.8) and (2.9), we have that Due to the Young inequality,

Lemma 2.4 We have that
In particular, assuming (1.6) and taking

Proof of Theorem 1.1
This section devoted to the proof of Theorem 1.1. 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.

Conflict of interest
The authors declare that they have no conflict of interest.
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