## Blow up for $$u_ t- \Delta u=g(u)$$ revisited.(English)Zbl 0855.35063

The authors consider the relation between the existence of global classical solutions of the equation $u_t- \Delta u= g(u)\text{ in } (0, \infty)\times \Omega,\;u= 0\text{ on } \partial\Omega,\;u(0)= u_0\text{ in } \Omega \tag{1}$ and the existence of weak solutions of the stationary problem (2) $$-\Delta u= g(u)$$ in $$\Omega$$, $$u= 0$$ on $$\partial\Omega$$. It is assumed (3) $$g: [0, \infty)\to [0, \infty)$$ is a $$C^1$$ convex nondecreasing function such that there exists $$x_0\geq 0$$ such that $$g(x_0)> 0$$ and $$\int_{x_0} ds/g(s)< \infty$$. The authors prove the following theorem: If a global classical solution of (1) exists for some $$u_0\in L^\infty(\Omega)$$, $$u_0\geq 0$$, then there exists a weak solution of (2). As a consequence it follows that if there is no weak solution of (2), then for any initial value $$u_0\in L^\infty(\Omega)$$, $$u_0\geq 0$$, the solution of (1) blows up in finite time. The authors also prove the following theorem without hypothesis (3): If there exists a weak solution $$w$$ of (2), then for any $$u_0\in L^\infty(\Omega)$$ with $$0\leq u_0\leq w$$, the solution of (1) with $$u(0)= u_0$$ is global.

### MSC:

 35K57 Reaction-diffusion equations 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

blow-up; stationary problem; global classical solution