zbMATH — the first resource for mathematics

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 \(r<p-1\).

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
Full Text: DOI