## Life span of solutions for a quasilinear parabolic equation with initial data having positive limit inferior at infinity.(English)Zbl 1282.35090

Blow up for the following Cauchy problem $u_{t} = \Delta {u^m} + {u^p},\,\,\,\, x \in\mathbb R^n,\,\,t > 0$
$u(x,0) = {u_0}(x) \geqslant 0,\,\,\, x \in\mathbb R^n,$ where $$1 < m < p,\,\,\, n \geqslant 1$$ and an initial datum $$u_0(x)$$ is a bounded continuous function on $$\mathbb R^n$$, is considered. It is proved the following estimates for blow up time.
Let $$n \geqslant 2.$$ Assume that there exist $$\xi \in {S^{n - 1}}$$ and $$\delta > 0$$ such that $\mathop {\text{ess}\inf }\limits_{\theta \in {S_\xi }(\delta )} \,{u_{0,\infty }}(\theta ) > 0.$ Then the weak solution blows up in finite time, and the blow-up time is estimated as ${T^*} \leqslant \frac{1}{{p - 1}}{\left( {\mathop {\text{ess}\inf }\limits_{\theta \in {S_\xi }(\delta )} \,{u_{0,\infty }}(\theta ) > 0} \right)^{1 - p}}.$ Let $$n=1.$$ Assume that $\max \left\{ \liminf_{x \to + \infty } {u_0}(x),\,\,\liminf_{x \to - \infty } {u_0}(x) \right\} > 0.$ Then the weak solution blows up in finite time and the blow-up time is estimated as ${T^*} \leqslant \frac{1}{{p - 1}}{\left( \max \left\{ \liminf_{x \to + \infty } {u_0}(x),\,\,\liminf_{x \to - \infty } {u_0}(x) \right\} \right)^{1 - p}}.$ Here for $$\xi \in {S^{n - 1}}$$ and $$\delta \in (0,\sqrt 2 )$$ ${\Gamma _\xi }(\delta ) = \left\{ {\eta \in {R^n}\backslash 0;\,\,\left| {\xi - \frac{\eta }{{\left| \eta \right|}}} \right| < \delta } \right\},$ and $${S_\xi }(\delta ) = {\Gamma _\xi }(\delta ) \cap {S^{n - 1}},$$ $${u_{0,\infty }}(\theta ) = \liminf_{r \to + \infty } {u_0}(r\theta )$$ for $$\theta \in {S^{n - 1}},$$ $${u_{0,\infty }} \in {L^\infty }({S^{n - 1}}).$$

### MSC:

 35B44 Blow-up in context of PDEs 35K15 Initial value problems for second-order parabolic equations 35K59 Quasilinear parabolic equations 35D30 Weak solutions to PDEs

### Keywords:

estimates for blow up time
Full Text:

### References:

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