论文标题
$ \ mathbb {r}^n $中的分数p-laplacian进化方程
The fractional p-Laplacian evolution equation in $\mathbb{R}^N$ in the sublinear case
论文作者
论文摘要
我们考虑自然依赖于时间的分数$ p $ -laplacian方程在整个欧几里得空间中，参数为$ 1 <p <2 $，而分数指数$ s \ in（0,1）$。相反，标准理论表明，列贝格$ l^q $空间中数据的凯奇问题是很好的，并且解决方案形成了一个具有规律性和其他有趣属性的非表达半群的家族。在最近的一篇论文中，已经处理了超线性案例$ p> 2 $。 我们在这里研究“快速”制度$ 1 <p <2 $，这更复杂。作为主要结果，我们为每个质量值$ m $构建自相似的基本解决方案，以及子量$ p_c = 2n/（n+s）<p <2 $中的任何$ p $，我们证明这是它们可以存在的确切范围。我们还证明，通用质量质量解决方案趋于具有相同质量的基本解决方案，并且融合在所有$ l^q $空间中都有。获得了全球Harnack不平等的形式的罚款。 本文的另一个主要主题是对具有强大奇异性的解决方案的研究。我们找到了一种称为“非常单数解决方案”的单数解决方案，该解决方案存在于$ p_c <p <p_1 $，其中$ p_1 $是我们引入的新临界数字，$ p_1 \ in（p_c，2）$。我们将这种类型的单数解决方案扩展到“非常快的范围” $ 1 <p <p_c $。它们代表了在较低的$ p $范围内具有有限时间灭绝的弱解决方案的例子。我们简要检查限制情况下的情况$ p = p_c $。最后，非常单数的解决方案与非线性特征值形式的分数椭圆问题有关。在限制情况下$ p = p_c $。最后，非常单数的溶液与非线性特征值形式的分数椭圆问题有关。
We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1<p<2$ and fractional exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $L^q$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case $p>2$ has been dealt with in a recent paper. We study here the "fast" regime $1<p<2$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass value $M$ and any $p$ in the subrange $p_c=2N/(N+s)<p<2$, and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all $L^q$ spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for $p_c<p<p_1$, where $p_1$ is a new critical number that we introduce, $p_1\in (p_c,2)$. We extend this type of singular solutions to the "very fast range" $1<p<p_c$. They represent examples of weak solutions having finite-time extinction in that lower $p$ range. We briefly examine the situation in the limit case $p=p_c$. Finally, very singular solutions are related to fractional elliptic problems of nonlinear eigenvalue form.in the limit case $p=p_c$. Finally, very singular solutions are related to fractional elliptic problems of nonlinear eigenvalue form.
