## 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 $$\beta= \sigma+1+ (\sigma+ 2)/N$$.

### MSC:

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

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