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Well-posedness for parabolic problems with variable-order nonlocal operators

Abstract:

In this paper, we study the existence and uniqueness of solutions to a parabolic-type problem in which the diffusion is governed by variable-order nonlocal operators, with kernels that depend on both space and time. Using viscosity solution techniques and considering a general class of nonlinearities characterized by gradient growth, we establish the existence of solutions via Perron’s method, as well as a comparison principle for bounded sub- and supersolutions. Furthermore, under a properness condition on the Hamiltonian, we analyze the long-time behavior of solutions and prove exponential convergence to the steady state.