Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators

Abstract: In this work, we address a parabolic problem featuring a potentially doubly nonlinear term, governed by a combination of local and nonlocal operators (see Problem P1 below). We first establish the local existence of weak energy solutions via a semidiscretization in time applied to an auxiliary evolution problem. The uniqueness of these solutions is subsequently obtained through a novel generalization of the classical inequality of Diaz and Saa, suitably adapted to the mixed local nonlocal setting. This generalization provides a new comparison principle and establishes the T-accretivity of a corresponding operator in L2. By employing this comparison principle, we construct suitable barrier functions that allow the global in time extension of solutions. Furthermore, we demonstrate the convergence of weak solutions to a nontrivial stationary state. Our approach relies on methods from the theory of contraction semigroups. It is noteworthy that these results are underpinned by a detailed analysis of the stationary problems associated with Problem P1, which also reveals several qualitative properties of the solutions.

• The paper establishes local existence and uniqueness of solutions for parabolic PDEs with mixed local and nonlocal p-Laplacians using advanced variational methods.
• It employs semi-discretization in time and convexity-based comparison principles to manage doubly nonlinear time derivatives and singular source terms.
• Regularity results include uniform boundary behavior and asymptotic convergence to a unique stationary state, ensuring stability and well-posedness.

Local Existence, Uniqueness, Regularity, and Global Behavior for Parabolic Equations with Mixed Local and Nonlocal Operators

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Introduction and Context

The study presented in "Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators" (2604.03931) is a rigorous and comprehensive contribution to the theory of quasilinear parabolic evolution equations characterized by the interplay between local (classical) and nonlocal (fractional) $p$-Laplacian operators. The analysis covers equations that can feature doubly-nonlinear time derivatives and may incorporate source terms with spatial singularities.

This research is motivated by the prominence of mixed dispersal processes in models across physics, biology (notably, population dynamics with both Brownian and Lévy-type dispersal), and material science. The mathematical analysis of PDEs with both local and nonlocal operators has matured for the stationary elliptic cases, but the understanding of parabolic, especially evolution, equations with nonlinear or doubly-nonlinear structure and source singularities is much less developed. The authors systematically address gaps in existence, uniqueness, regularity, and long-term behavior of weak solutions to prototypical problems of the form

$$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$

with Dirichlet-type boundary conditions, where the second and third terms represent the local and nonlocal $p$-Laplacians, respectively, and $d(x)$ is the distance to the boundary.

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Main Analytical Framework and Results

Functional Setting and Weak Formulation

The authors develop a robust functional analytic setting, working on $L^\infty$-in-time and Sobolev-in-space frameworks, and precisely define the relevant spaces, norm equivalences, and embedding properties necessary for dealing with critical and supercritical nonlinearities, including those arising from the singular lower-order terms. The definition of weak solution incorporates the nontrivial time-derivative term, which, for $m>0$, leads to a doubly-nonlinear structure in time.

A significant technical point is the use of semi-discretization in time (Rothe's method), which allows for a successful passage from discrete-in-time problems—handled via energy minimization and variational techniques for an implicit Euler scheme—to solutions to the continuous-time PDE.

Existence and Uniqueness

The local existence of energy solutions is established via time semi-discretization: at each time step, a nonlinear elliptic Dirichlet problem with mixed operators is solved by minimization of a coercive and weakly lower semi-continuous energy functional, whose structure is closely related to the underlying PDE. Coercivity and regularizing effects are demonstrated even in the presence of singular lower order potentials and source terms involving distance-to-boundary weights.

A new comparison principle for sub- and supersolutions is derived via a generalization of the Díaz–Saa inequality to the mixed local–nonlocal context. The key technical result is a convexity property for specific functionals on cones in function spaces, allowing strict monotonicity and a $T$-accretivity property of the associated nonlinear operator in $L^2$, which is then leveraged to deduce uniqueness of weak solutions for any $m \ge 0$.

Notably, the uniqueness remains valid without imposing the smallness condition $p(1-s)<1$ often present in previous works on related equations.

Regularity and Asymptotic Behavior

The constructed solutions enjoy fine regularity: for appropriate parameters (including $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$0, $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$1, and restrictions on the exponents in the nonlinearities and singular terms), $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$2 with uniform, time-independent estimates in the form

$$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$3

for almost every $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$4 in the domain, where $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$5 is explicitly determined by the parameters $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$6 and the regularity of the boundary singularity. This type of estimate is optimal, respecting the intrinsic boundary behavior governed by the mixed local-nonlocal operator.

Energy identities and contractivity estimates are proven, quantifying both the regularization and the stability of the dynamics. Using contraction semigroup theory, solutions are shown to converge, as $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$7, towards a unique, nontrivial stationary state—the stationary solution of the associated elliptic mixed problem with the limit potential.

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Qualitative Features and Innovative Technical Contributions

Several aspects of the analysis are of particular interest:

• Strict convexity and Picone-type inequalities for mixed operators: The extension of variational tools (convex energies, Picone identities, Díaz–Saa type inequalities) to the setting of coupled local and nonlocal, possibly degenerate, quasilinear operators is delicate due to the lack of direct maximum principle analogues in the nonlocal context. These techniques underpin the comparison and monotonicity principles for weak solutions.
• Simultaneous control of local and nonlocal terms: All regularity and stability analysis require controlling both the classical and fractional gradient energies; the proofs delicately balance the coercivity and compactness imparted by each.
• Handling of double nonlinearity in time: For $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$8, the time evolution is not governed by the standard $$\partial_t \left(u^{2m+1}\right) - \Delta_p u + (-\Delta)_p^s u = g(t,x)u^m + m d(x)^{-\gamma} u^\delta,$$9, but by a more general $p$0 or, equivalently, a doubly nonlinear evolution in $p$1. The analytic approach manages the transfer of nonlinearity between the time-derivative and the energy terms to establish strong a priori bounds and continuous dependence on initial data.

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Implications and Prospects

This work solidifies the foundations for the analysis of quasilinear parabolic PDEs with local–nonlocal structure. The main implications include:

• Flexibility for applications: The results, robust against strong nonlinearities and singularities, are directly applicable to mathematical models in heterogeneous diffusion (e.g., combining local exploration and long-jump processes), and more generally, to problems in which both local and nonlocal spatial effects must be simultaneously accounted for.
• Analytical paradigm for further studies: The techniques—especially the comparison/contraction principles and the semigroup approach—provide a template for analyzing related doubly-nonlinear, degenerate, or singular evolution equations with even more complex operator combinations or boundary behaviors.
• Foundations for numerical analysis and control: The developed theory ensures well-posedness and regularity up to the boundary, enabling rigorous analysis of numerical schemes (leveraging the discrete-in-time theory) and supporting future work in optimal control or inverse problems for such evolution equations.
• Potential for extension to vector-valued and system cases: The abstract functional analytic machinery and variational principle extensions may be adapted to systems of coupled local–nonlocal equations, degenerate operators with spatially varying exponents, or nontrivial time-dependent operator coefficients.

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Conclusion

The paper provides a thorough, technically rigorous treatment of the key analytical issues for mixed local–nonlocal, quasilinear parabolic evolution equations, including existence, uniqueness, boundary regularity, and large-time asymptotics. The developed tools—notably the convexity-based comparison principle and the contraction semigroup framework—resolve long-standing obstacles in this emerging research area and lay the groundwork for further theoretical, numerical, and applied developments in mixed operator parabolic PDEs (2604.03931).