Blow-up for quasilinear heat equations with critical Fujita’s exponents. (English) Zbl 0808.35053

Summary: We consider the Cauchy problem for the quasilinear heat equation \[ u_ t= \text{div} (u^ \sigma \nabla u)+ u^ \beta \quad \text{for } x\in\mathbb{R}^ N, \quad t>0, \] where \(\sigma>0\) is a fixed constant, with the critical exponent in the source term \(\beta= \beta_ c= \sigma+1+ 2/N\). It is well-known that if \(\beta\in (1,\beta_ c)\) then any nonnegative weak solution \(u(x,t)\not\equiv 0\) blows up in a finite time.
In the present paper we prove that \(u\not\equiv 0\) blows up in the critical case \(\beta= \sigma+1+ 2/N\) with \(\sigma>0\). A similar result is valid for the equation with gradient-dependent diffusivity \[ u_ t= \text{div} (| Du|^ \sigma Du)+ u^ \beta, \] with \(\sigma>0\), and the critical exponent \(\beta= \sigma+1+ (\sigma+ 2)/N\).


35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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