Differential and Integral Equations
- Differential Integral Equations
- Volume 20, Number 9 (2007), 1021-1034.
On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent
In this paper, we are concerned with the existence of $L^\infty$-global bounds for global-in-time solutions of some semilinear parabolic problems. It is well known that every global-in-time solution for the subcritical problem is globally bounded in $L^\infty$, while there exists a global solution which is not bounded in $L^\infty$ globally in time in the critical case. In this paper, we discuss the necessary and sufficient condition for the existence of $L^\infty$-global bounds, which is valid for the subcritical and the critical case in a unified way. Moreover, using our main results, we provide various examples with the critical exponent in which every global-in-time solution has an $L^\infty$-global bound.
Differential Integral Equations Volume 20, Number 9 (2007), 1021-1034.
First available in Project Euclid: 20 December 2012
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Ishiwata, Michinori. On bounds for global solutions of semilinear parabolic equations with critical and subcritical Sobolev exponent. Differential Integral Equations 20 (2007), no. 9, 1021--1034. https://projecteuclid.org/euclid.die/1356039309.