Stability of planar diffusion wave for nonlinear evolution equation

Abstract

It is known that the one-dimensional nonlinear heat equation \(u_t = f(u)_{x_1 x_1 }\), f′(u) > 0, u(±∞, t) = u−, u+ ≠ u_ has a unique self-similar solution \(\bar u\left( {\tfrac{{x_1 }} {{\sqrt {1 + t} }}} \right)\). In multi-dimensional space, \(\bar u\left( {\tfrac{{x_1 }} {{\sqrt {1 + t} }}} \right)\) is called a planar diffusion wave. In the first part of the present paper, it is shown that under some smallness conditions, such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation: ut − Δf(u) = 0, x ∈ ℝn. The optimal time decay rate is obtained. In the second part of this paper, it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping: utt + ut − Δf(u) = 0, x ∈ ℝn. The time decay rate is also obtained. The proofs are given by an elementary energy method.