关于分数阶1-拉普拉斯发展方程
On fractional $1$-Laplacian evolution equation
AI总结：
该研究针对带Carathéodory非线性项的分数阶1-拉普拉斯非局部发展问题，通过位势井框架、Galerkin逼近等方法，建立弱解与强解的整体存在性及局部存在性，探究解的定性行为。
AI中文摘要：
我们研究由分数阶1-拉普拉斯驱动、带有满足次临界增长条件的Carathéodory非线性项的非局部发展问题。我们建立位势井框架，以探究不同初始能量水平下解的存在性与定性行为。通过结合改进的位势井方法与Galerkin逼近，我们在初始能量的适当条件及相关Nehari泛号的符号下，建立了弱解与强解的整体存在性。为处理分数阶1-拉普拉斯的奇异性，我们用一族分数阶p-拉普拉斯方程逼近该问题，推导关于p>1的一致估计，并取p→1+的极限。此外，在低维区域N<2s中，我们采用次微分方法建立强解的局部存在性并研究其定性行为。
英文摘要：
We study a nonlocal evolution problem driven by the fractional $1$-Laplacian with a Carathéodory nonlinearity satisfying subcritical growth conditions. We develop a potential well framework to investigate the existence and qualitative behavior of solutions at different initial energy levels. By combining a modified potential well method with Galerkin approximations, we establish the global existence of weak and strong solutions under appropriate conditions on the initial energy and the sign of the associated Nehari functional. To address the singular nature of the fractional $1$-Laplacian, we approximate the problem by a family of fractional $p$-Laplacian equations, derive estimates uniform with respect to $p>1$, and pass to the limit as $p\to 1^{+}$. Furthermore, in the low-dimensional regime $N<2s$, we employ a subdifferential approach to establish the local existence of strong solutions and investigate their qualitative behavior.
