On the Cauchy problem for the Satsuma-Mimura diffusion equation. (English) Zbl 0689.45014
The author considers the initial value problem \(u_ t-au_{xx}+b[S_ u\cdot u]_ x=0,\quad u(x,0)=f(x).\) f(x) is assumed to be Hölder continuous function. With the use of perturbation methods, and complex analysis, this problem is transformed to simple Hammerstein integral equation.
The author proves the existence of a solution u(x,t) to the problem which is continuous, bounded, and satisfies a certain minimum principle.
The author proves the existence of a solution u(x,t) to the problem which is continuous, bounded, and satisfies a certain minimum principle.
Reviewer: H.S.Nur
Keywords:
Cauchy problem; Satsuma-Mimura diffusion equation; initial value problem; perturbation methods; Hammerstein integral equation; existence; minimum principleReferences:
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