Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption
Vol. 17 (2010), No. 4, Page 323--358.
Galaktionov, V. A.
Vast Multiplicity of Very Singular Self-Similar Solutions of a Semilinear Higher-Order Diffusion Equation with Time-Dependent Absorption
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Abstract:
As a basic model, the Cauchy problem in $\ren \times \re_+$ for the $2m$th-order semilinear parabolic equation of the diffusion-absorp\-tion type $$ u_t =-(- \D)^m u - t^\a |u|^{p-1}u, \quad \mbox{with} \,\, p > 1, \,\, \a > 0, \,\,\, m \ge 2, $$ with {\em singular} initial data $u_0(x) \not \equiv 0$ such that $ u_0(x)=0$ for any $x \not = 0$, is studied. The additional multiplier $h(t)=t^\a \to 0$ as $t \to 0$ 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 {\em very singular solutions} (VSSs) is studied. For $m=1$ and $h(t) \equiv 1$, first nonexistence result for $p \ge p_0=1+\frac 2N$ was proved in the celebrated paper by Brezis and Friedman in 1983. Existence of VSSs in the complement interval $1
Keywords: The Cauchy problem, diffusion equations with absorption, initial Dirac mass, very singular solutions, existence, nonexistence, bifurcations, branching.
Mathematics Subject Classification (2010): Primary 35K55, Secondary 35K40, 35K65.
Received: 2007-11-05
