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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
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References:

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