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Global stability of traveling waves for nonlocal time-delayed degenerate diffusion equation. (English) Zbl 1478.35039

Summary: This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Hölder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally \(L^1\)-stable, which is the first frame work on \(L^1\)-wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical \(L^1\)-weighted energy estimates combining the compactness analysis, but with some new development.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35C07 Traveling wave solutions
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35K59 Quasilinear parabolic equations
35R09 Integro-partial differential equations
35R10 Partial functional-differential equations
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