Abstract
In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].
Highlights
This paper deals with the following classical semilinear parabolic equation associated with critical Sobolev exponent in RN: ut − ∆u = |u|p−1u in RN × (0, T), u|t=0 = u0(x) in RN, (1.1)
There is a great literature on the existence of global solutions and blow-up for the problem (1.1) on the bounded domain:
It is well known that there exist choices of u0 for which the corresponding solutions tend to zero as t → ∞ and other choices for which the solutions blow-up in finite time
Summary
After the pioneer work of Sattinger and Payne, some authors [7, 9,10,11,12, 17,18,19, 26] used the method to study the global existence and nonexistence of solutions for various nonlinear evolution equations with initial boundary value problem. We use the modified potential well method to obtain global existence and blow up in finite time of solutions when the initial energy is low, critical and high, respectively.
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