Abstract
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.
Highlights
In this paper, we investigate the well-posedness of the following Cauchy problem:∂t u + κ (∂ x u)2 + q(∂ x u)3 + r (∂ x u)4 + δ∂3x u+ β2 ∂4x u + μ∂2x uε + γu∂2x u = 0, u(0, x ) = u ( x ), t > 0, x ∈ R, (1)x ∈ R, with γ = 0, β 6= 0, q = r = 0, or β 6= 0, (2) κ = 2γ. (3)Under Assumption (2), we assume on the initial datum u0 ∈ H 2 (R). (4)Instead, under Assumption (3), we assume (4) or u0 ∈ H 3 (R)
(4), Theorem 1 gives only the existence of the solution, while the uniqueness is guaranteed by Assumption (5)
The main result of this paper is the following theorem
Summary
We investigate the well-posedness of the following Cauchy problem: Observe that thanks to (7), Equation (11) is equivalent to the following one 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.
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