Global behavior of solutions of a reaction-diffusion equation with gradient absorption in unbounded domains
Global behavior of solutions of a reaction-diffusion equation with gradient absorption in unbounded domains
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发表时间：
2006
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通讯作者：
Jean-Philippe Bartier
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作者：
Jean-Philippe Bartier
我们研究了方程∂tu−Δu=up - μ |∇u|qur在一般域（有界或无界）。在r=0的情况下建立了许多结果，我们将推广一些性质。如果q≥1且q+r≥p，我们将证明Ω（或庞加莱不等式）内半径的有限性与解的整体存在性之间有很强的联系。更准确地说，如果Ω的内半径是有限的，那么解u是全局的，并且如果μ足够大，解呈指数衰减到零。反之，如果它是无限的，则总是存在无界解。还得到了其他定性结果。
We study the equation ∂tu−Δu=up−μ |∇u|qur in a general domain (bounded or not). Many results have been established in the case r=0, and we will generalize some properties. If q≥1 and q+r≥p, we will show that there is a strong connection between the finiteness of the inradius of Ω (or the Poincare inequality) and the global existence of the solutions. More precisely if the inradius of Ω is finite, then the solutions u are global and if moreover μ is large enough, the solutions decay exponentially to zero. Conversely, if it is infinite, there always exists unbounded solution. Other qualitative results are also obtained.