Recent results and open problems on parabolic equations with gradient nonlinearities.
Recent results and open problems on parabolic equations with gradient nonlinearities.
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发表时间：
2001
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通讯作者：
P. Souplet
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作者：
P. Souplet
我们调查了最近的结果，并提出了许多有关具有梯度非线性的半线性抛物型方程解的长期行为的开放问题。我们关注具有耗散梯度项的模型方程 ut u = u p bjruj q ；其中 p、q > 1、b > 0，具有齐次狄利克雷边界条件。在过去的十年里，有许多论文致力于这个方程，我们将结果与纯功率反应扩散方程（b = 0）情况下的已知结果进行比较。在存在耗散梯度项的情况下，会出现一些当 b = 0 时不会出现的新现象。所处理的问题涉及：爆炸的充分条件、爆炸解的行为、全局存在性和稳定性、无界全局解、临界指数和稳态。
We survey recent results and present a number of open problems concerning the large-time behavior of solutions of semilinear parabolic equations with gradient nonlinearities. We focus on the model equation with a dissipative gradient term ut u = u p bjruj q ; where p, q > 1, b > 0, with homogeneous Dirichlet boundary conditions. Numerous papers were devoted to this equation in the last ten years, and we compare the results with those known for the case of the pure power reaction-diusion equation ( b = 0). In presence of the dissipative gradient term a number of new phenomena appear which do not occur when b = 0. The questions treated concern: sucient conditions for blowup, behavior of blowing up solutions, global existence and stability, unbounded global solutions, critical exponents, and stationary states.