ABSTRACT This paper is concerned with the well‐posedness and existence of weak pullback mean random attractors for a class of reaction–diffusion equations with nonlocal coefficient and nonlinear multiplicative noise. In our problem, the coefficient can be degenerate, and the nonlinearity is not locally Lipschitz on the phase space.

Abstract In this paper, we demonstrate that the so-called expansion of positivity holds true for doubly nonlinear nonlocal parabolic type equations, having the fractional p-Laplace type operator and a power-nonlinearity in the time-derivative. The exponential time-stretching method originally developed for the local equations is transparently extended to the nonlocal equations in the scaling regime intrinsic to the doubly nonlinear nonlocal parabolic operator. The nonlocal effect of the nonlocal equations are given by the so-called tail.

Weighted Sobolev space theory for non-local elliptic and parabolic equations with nonzero exterior condition on $$C^{1,1}$$ open sets

Non-local Parabolic Equations with Singular (Morrey) Time-Inhomogeneous Drift

We use the method of moving planes to prove the radial symmetry and monotonicity of solutions of fractional parabolic equations in the unit ball. Since the fractional Laplacian operator is a linear operator, we investigate the maximal regularity of nonlocal parabolic fractional Laplacian equations in the unit ball. The maximal regularity of nonlocal parabolic fractional Laplacian equations guarantees the existence of solutions in the unit ball. Based on these conditions, we first establish a maximum principle in a parabolic cylinder, and the principles provide a starting position to apply the method of moving planes. Then, we consider the fractional parabolic equations and derive the radial symmetry and monotonicity of solutions in the unit ball.

Pointwise and Oscillation Estimates Via Riesz Potentials for Mixed Local and Nonlocal Parabolic Equations

Abstract We establish new Harnack estimates that defy the waiting‐time phenomenon for global solutions to nonlocal parabolic equations. Our technique allows us to consider general nonlocal operators with bounded measurable coefficients. Moreover, we show that a waiting‐time is required for the nonlocal parabolic Harnack inequality when local solutions are considered.

Abstract In this paper, we study a nonlinear system involving a generalized tempered fractional p-Laplacian in B 1(0): ∂ t u ( x , t ) + ( − Δ − λ f ) p s u ( x , t ) = g ( t , u ( x , t ) ) , ( x , t ) ∈ B 1 ( 0 ) × [ 0 , + ∞ ) , u ( x ) = 0 , ( x , t ) ∈ B 1 c ( 0 ) × [ 0 , + ∞ ) , $$\begin{cases}_{t}u\left(x,t\right)+{\left(-{\Delta}-{\lambda }_{f}\right)}_{p}^{s}u\left(x,t\right)=g\left(t,u\left(x,t\right)\right),\quad \hfill & \left(x,t\right)\in {B}_{1}\left(0\right){\times}\left[0,+\infty \right),\hfill \\ u\left(x\right)=0,\quad \hfill & \left(x,t\right)\in {B}_{1}^{c}\left(0\right){\times}\left[0,+\infty \right),\hfill \end{cases}$$ where 0 < s < 1, p > 2, n ≥ 2. We establish Hopf’s lemma for parabolic equations involving a generalized tempered fractional p-Laplacian. Hopf’s lemma will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.

In this work, we obtain quantitative estimates of the continuity constant for the \(L^p\) maximal regularity of relatively continuous nonautonomous operators \(A : I \to \mathcal{L}{L}(D,X)\), where \(D \hookrightarrow X\) densely and compactly. They allow in particular to establish a new general growth condition for the global existence of strong solutions of Cauchy problems for nonlocal quasilinear equations for a certain class of nonlinearities \(u \mapsto \mathbb{A}(u)\). The estimates obtained rely on the precise asymptotic analysis of the continuity constant with respect to perturbations of the operator of the form \(A(\cdot) + \lambda \text{Id}\) as \(\lambda \to \pm \infty\). A complementary work in preparation supplements this abstract inquiry with an application of these results to nonlocal parabolic equations in noncylindrical domains depending on the time variable. For more information see https://ejde.math.txstate.edu/Volumes/2025/18/abstr.html

The regularity properties of nonlocal anisotropic elliptic equations with parameters are investigated in abstract weighted Lp spaces. The equations include the variable coefficients and abstract operator function A = A (x) in a Banach space E in leading part. We find the sufficient growth assumptions on A and appropriate symbol polynomial functions that guarantee the uniformly separability of the linear problem. It is proved that the corresponding anisotropic elliptic operator is sectorial and is also the negative generator of an analytic semigroup. Byusing these results, the existence and uniqueness of maximal regular solution of the nonlinear nonlocal anisotropic elliptic equation is obtained in weighted Lp spaces. In application, the maximal regularity properties of the Cauchy problem for degenerate abstract anisotropic parabolic equation in mixed Lp norms, the boundary value problem for anisotropic elliptic convolution equation, the Wentzel-Robin type boundary value problem for degenerate integro-differential equation and infinite systems of degenerate elliptic integro-differential equations are obtained.

Nonlocal Separable Elliptic and Parabolic Equations and Applications

A priori bounds and ground states of nonlocal parabolic equations

Long time behavior of a class of nonlocal parabolic equations without uniqueness

On the nonexistence of global solutions for nonlocal parabolic equations with forcing terms

We prove optimal regularity results in Lp-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a (0<a<1) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian (-Δ)a, as well as elliptic operators (-∇·A(x)∇+b(x))a. The proofs are based on general results on maximal Lp-regularity and its relation to R-boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Abstract A general modulus of continuity is quantified for locally bounded, local, weak solutions to nonlocal parabolic equations, under a minimal tail condition. Hölder modulus of continuity is then deduced under a slightly stronger tail condition. These regularity estimates are demonstrated under the framework of nonlocal ‐Laplacian with measurable kernels.

We investigate the local properties, including the nodal set and the nodal properties of solutions to the following parabolic problem of Muckenhoupt-Neumann type: { ∂ t u ¯ − y − a ∇ ⋅ ( y a ∇ u ¯ ) = 0 a m p ; in B 1 + × ( − 1 , 0 ) − ∂ y a u ¯ = q ( x , t ) u a m p ; on B 1 × { 0 } × ( − 1 , 0 ) , \begin{equation*} \begin {cases} \partial _t \overline {u} - y^{-a} \nabla \cdot (y^a \nabla \overline {u}) = 0 \quad &amp;\text { in } \mathbb {B}_1^+ \times (-1,0) \\ -\partial _y^a \overline {u} = q(x,t)u \quad &amp;\text { on } B_1 \times \{0\} \times (-1,0), \end{cases} \end{equation*} where a ∈ ( − 1 , 1 ) a\in (-1,1) is a fixed parameter, B 1 + ⊂ R N + 1 \mathbb {B}_1^+\subset \mathbb {R}^{N+1} is the upper unit half ball and B 1 B_1 is the unit ball in R N \mathbb {R}^N . Our main motivation comes from its relation with a class of nonlocal parabolic equations involving the fractional power of the heat operator H s u ( x , t ) = 1 | Γ ( − s ) | ∫ − ∞ t ∫ R N [ u ( x , t ) − u ( z , τ ) ] G N ( x − z , t − τ ) ( t − τ ) 1 + s d z d τ . \begin{equation*} H^su(x,t) = \frac {1}{|\Gamma (-s)|} \int _{-\infty }^t \int _{\mathbb {R}^N} \left [u(x,t) - u(z,\tau )\right ] \frac {G_N(x-z,t-\tau )}{(t-\tau )^{1+s}} dzd\tau . \end{equation*} We characterise the possible blow-ups and we examine the structure of the nodal set of solutions vanishing with a finite order. More precisely, we prove that the nodal set has at least parabolic Hausdorff codimension one in R N × R \mathbb {R}^N\times \mathbb {R} , and can be written as the union of a locally smooth part and a singular part, which turns out to possess remarkable stratification properties. Moreover, the asymptotic behaviour of general solutions near their nodal points is classified in terms of a class of explicit polynomials of Hermite and Laguerre type, obtained as eigenfunctions to an Ornstein-Uhlenbeck type operator. Our main results are obtained through a fine blow-up analysis which relies on the monotonicity of an Almgren-Poon type quotient and some new Liouville type results for parabolic equations, combined with more classical results including Federer’s reduction principle and the parabolic Whitney’s extension.

We study nonlocal elliptic and parabolic equations on C 1,τ open sets in weighted Sobolev spaces, where τ ∈ (0, 1). The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable.

We consider mixed local and nonlocal quasilinear parabolic equations of p-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions, local Holder continuity of weak solutions, lower semicontinuity of weak supersolutions as well as upper semicontinuity of weak subsolutions. We also discuss the pointwise behavior of the semicontinuous representatives. Our main results are valid for sign changing solutions. Our approach is purely analytic and is based on energy estimates and the De Giorgi theory.

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.