论文标题
具有单数和非十个初始数据的分数半线性热方程
Fractional semilinear heat equations with singular and nondecaying initial data
论文作者
论文摘要
我们研究了分数半线性热方程的局部积分解决方案的存在和不存在的可集成性条件，而在统一的本地$ l^p $空间中，具有相当普遍的非线性。我们有关此问题的主要结果由定理1.4、1.6、5.1和5.3组成。我们引入了一种新的超级分解，起着至关重要的作用。我们的方法不依赖变量的变化，因此可以应用于一类广泛的非局部抛物线方程。特别是，当非线性术语为$ u^p $或$ e^u $时，可以在关键案例中构建局部解决方案，并且存在和不存在的不存在的整合条件被完全分类。我们的分析基于比较原则，詹森的不平等和$ l^p $ - $ l^q $类型估计。
We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $L^p$ spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a new supersolution which plays a crucial role. Our method does not rely on a change of variables, and hence it can be applied to a wide class of nonlocal parabolic equations. In particular, when the nonlinear term is $u^p$ or $e^u$, a local-in-time solution can be constructed in the critical case, and integrability conditions for the existence and nonexistence are completely classified. Our analysis is based on the comparison principle, Jensen's inequality and $L^p$-$L^q$ type estimates.
