## Blow-up of positive solutions of semilinear heat equations.(English)Zbl 0576.35068

The following problem is discussed:
$$u_ t=\Delta u+f(u)$$ in $$\Omega$$ $$\times (0,T)$$, $$f\in C^ 1$$, $$f(s)>0$$ if $$s>0$$, $$\Omega \subset R^ n$$; $$u(x,0)=\phi (x)$$ if $$x\in \Omega$$, $$\phi \in C^ 1({\bar \Omega})$$, $$\phi\geq 0$$, $$\phi =0$$ on $$\partial \Omega$$, $$u(x,t)=0$$, $$x\in \partial \Omega$$, $$0<t<T.$$
As $$U(t)=\max_{x\in \Omega} u(x,t)$$ grows with t, it is assumed that $$T<\infty$$ is the supremum of all $$\sigma$$ such that the solution to the problem above exists for $$t<\sigma$$, and that $$U(T-)=+\infty:x\in \Omega$$ is a blow-up point if there is $$\{(x_ mt_ m)\}$$, $$t_ m\uparrow T$$, $$x_ m\to X$$, and $$u(x_ m,t_ m)\to \infty$$, $$m\to \infty.$$
A partial outline of the results can be divided into:
Case (i): $$\Omega$$ is a ball, u(.,t) are radial functions, $$\phi_ r\leq 0$$. The authors prove that the only blow-up point is $$x=0$$. For the case $$f(u)=(u+\lambda)^ p$$, $$p>1$$, $$\lambda\geq 0$$, they obtain: $$| u(r,t)| \leq C/r^{2/(\lambda -1)}$$, any $$\gamma <p$$, $$\lim_{t\to T} \sup \| u(.,t)\|_{L^ q(\Omega)}<\infty$$, $$q<n(p-1)/2$$, $$\lim_{t\to T} \inf \| u(.,t)\|_{L^ q(\Omega)}=\infty$$, $$q>n(p-1)/2$$; if moreover $$\Delta \phi +f(\phi)\geq 0$$ and $$n=1,2$$; or $$n\geq 3$$ and $$p\leq (n+2)/n-2$$, $$(T-t)^{1/(p-1)}u(r,t)\to (1/p- 1)^{1/p-1}$$ as $$t\uparrow T$$ provided $$r\leq C(T-t)^{1/2}$$, some $$C>0.$$
Case (ii): Non symmetric, $$\Omega$$ is a convex domain: the blow-up points lie in a compact subset of $$\Omega$$. For the particular $$f(u)=(u+\lambda)^ p$$, u(x,t)$$\leq^ C/(T-t)^{1/(p-1)}$$, all $$x\in \Omega$$, and $$\lim_{t\to T} \inf \| u(.,t)\|_{L^ q(\Omega)}=\infty$$ if $$q>n(p-1)/2.$$
In the one-dimensional case $$n=1$$, if $$\phi$$ ’ changes sign just once, then the solution blow-up at a single point. The authors extend some of the results to the case of a boundary condition $$\partial u/\partial \nu +\beta u=0$$, $$x\in \partial \Omega$$, $$0<t<T$$, $$\beta\geq 0$$ and $$\nu$$ the outward normal vector.
Reviewer: J.E.Bouillet

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

### Keywords:

positive solutions; semilinear heat equations; blow-up
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