Global existence and blow-up for semilinear parabolic equation with critical exponent in RN

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 RN. 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 L2 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. Poincare Anal. Non Lineaire 27(2010), No. 3, 877- 900].

Automated summary

This paper uses 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 RN. Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. The paper extends recent results obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincare Anal. Non Lineaire 27(2010), No. 3, 877- 900].