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Finite time vs. infinite time gradient blow-up in a degenerate diffusion equation. (English) Zbl 1169.35035
Summary: This paper deals with the phenomenon of gradient blow-up of nonnegative classical solutions of the Dirichlet problem for
$u_t= u^p u_{xx}+ \kappa u^r u_x^2+ u^q \quad\text{in }\Omega\times(0,T) \tag $$*$$$
in a bounded interval $$\Omega\subset\mathbb R$$, where $$p>2$$, $$1\leq q\leq p-1$$, $$r\geq 1$$, $$\kappa\geq 0$$, and the initial data $$u_0$$ are assumed to belong to $$C^1(\overline{\Omega})$$ and satisfy $$u_0(x)\geq c_0 \operatorname{dist}(x,\partial\Omega)$$ in $$\Omega$$ with some $$c_0=0$$. It is shown that if the gradient term in $$(*)$$ is weak enough near $$u=0$$, then all bounded solutions undergo an infinite time gradient blow-up, whereas if this term is sufficiently strong, then all solutions blow up in $$C^1(\overline{\Omega})$$ within finite time. Here by ‘weak’ we mean that the parameters satisfy either $$r=p-1$$ and $$\kappa\leq p-2$$, or $$r>p-1$$, and by strong the precise opposite, that is, either $$r=p-1$$ and $$\kappa>p-2$$, or $$r<p-1$$.

##### MSC:
 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations 35B33 Critical exponents in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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