Vast multiplicity of very singular self-similar solutions of a semilinear higher-order diffusion equation with time-dependent absorption

Reference:

Galaktionov, V. A., 2010. Vast multiplicity of very singular self-similar solutions of a semilinear higher-order diffusion equation with time-dependent absorption. Journal of Mathematical Sciences-the University of Tokyo, 17 (4), pp. 323-358.

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Abstract

As a basic model, the Cauchy problem in R-N x R+ for the 2mth-order semilinear parabolic equation of the diffusion-absorption type u(t) = -(-Delta)(m)u - t(alpha)vertical bar u vertical bar(p-1)u, with p > 1, alpha > 0, m >= 2, with singular initial data u(0)(x) not equivalent to 0 such that u(0)(x) = 0 for any x not equal 0, is studied. The additional multiplier h(t) = t(alpha) -> 0 as t -> 0 in the absorption term plays a role of a time-dependent non-homogeneous potential that affects the strength of the absorption term in the PDE. Existence and nonexistence of the corresponding very singular solutions (VSSs) is studied. For m = 1 and h(t) 1, first nonexistence result for p >= p(0) = 1+ 2/N was proved in the celebrated paper by Brezis and Friedman in 1983. Existence of VSSs in the complement interval 1 < p < p(0) was established in the middle of the 1980s. The main goal is to justify that, in the subcritical range 1 < p < p(0) = 1 + 2m(1+alpha)/N, there exists a finite number of different VSSs of the self-similar form U-*(x, t) = t(-beta)V(y), y = x/t(1/2m), beta = 1+alpha/p-1, where each V is an exponentially decaying as y -> infinity solution of the elliptic equation -(-Delta)V-m + 1/2m y center dot del V + beta V - vertical bar V vertical bar(p-1) V=0 in R-N. Complicated families of VSSs in 1D and also non-radial VSS patterns in RN are detected. Some of these VSS profiles V-l are shown to bifurcate from 0 at the bifurcation exponents p(l) = 1 + 2m(1+alpha)/l+N where l = 0, 1, 2, ... .

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