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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月30日

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  • Miguel Yangari,
  • Giovanella Peña

摘要

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.