Abstract We consider notions of weak solutions to a general class of parabolic problems of linear growth, formulated independently of time regularity. Equivalence with variational solutions is established using a stability result for weak solutions. A key tool in our arguments is approximation of parabolic BV functions using time mollification and Sobolev approximations. We also prove a comparison principle and a local boundedness result for solutions. When the time derivative of the solution is in $$L^2$$ L 2 our definitions are equivalent with the definition based on the Anzellotti pairing.

Abstract We study a reaction–diffusion partial differential equation (PDE) system with a distributed input, subject to multiple unknown plant parameters with arbitrarily large uncertainties. Using Lyapunov-based techniques, we design a delay-adaptive predictor feedback controller that ensures local boundedness of system trajectories and asymptotic regulation of the closed-loop system in terms of the plant state. Specifically, we model the input delay as a one-dimensional transport PDE with a spatial variable, effectively transforming the time delay into a spatially distributed shift. For the resulting coupled transport and reaction–advection–diffusion PDE system, we employ a PDE backstepping approach combined with the certainty-equivalence principle to derive an adaptive control law that compensates for both the unknown time delay and the unknown functional parameters. Simulation results are provided to illustrate the feasibility of our control design.

Abstract We investigate quasi-symmetry for small perturbations of the Gidas-Ni-Nirenberg problem involving the p -Laplacian and for small perturbations the critical p -Laplace equation for $$p>2$$ p > 2 . To achieve these results, we provide a quantitative review of the work by Damascelli & Sciunzi [16] concerning the weak Harnack comparison inequality and the local boundedness comparison inequality. Moreover, we prove a comparison principle for small domains.

In this article, using the De Giorgi-Nash-Moser method, we obtain an interior Holder continuity of weak solutions to nonlocal \(p\)-Laplacian type Schrodinger equations given by an integro-differential operator \(L^p_K\) (\(p >1\)), $$\displaylines{ L^p_K u+V|u|^{p-2} u=0 \quad\text{in } \Omega, \cr u=g \quad \text{in } \mathbb{R}^n\backslash \Omega. }$$ Where \(V=V_+-V_-\) with \((V_-,V_+)\in L^1_{loc}(\mathbb{R}^n)\times L^q_{loc}(\mathbb{R}^n)\) for \(q >\frac{n}{ps} >1\) and \(0< s< 1\) is a potential such that \((V_-,V_+^{b,i})\) belongs to the \((A_1,A_1)\)-Muckenhoupt class and \(V_+^{b,i}\) is in the \(A_1\)-Muckenhoupt class for all \(i\in\mathbb{}N\). Here, \(V_+^{b,i}:=V_+\max\{b,1/i\}/b\) for an almost everywhere positive bounded function \(b\) on \(\mathbb{R}^n\) with \(V_+/b\in L^q_{ loc}(\mathbb{R}^n)\), \(g\in W^{s,p}(\mathbb{R}^n)\) and \(\Omega\subset\mathbb{R}^n\) is a bounded domain with Lipschitz boundary. In addition, we prove local boundedness of weak subsolutions of the nonlocal \(p\)-Laplacian type Schrodinger equations. Also we obtain the logarithmic estimate of the weak supersolutions which play a crucial role in proving Holder regularity of the weak solutions. We note that all the above results also work for a nonnegative potential in \(L^q_{loc}(\mathbb{R}^n)\) (\(q >\frac{n}{ps} >1\), \(0< s< 1\)). For more information see https://ejde.math.txstate.edu/Volumes/2025/83/abstr.html

Abstract We extend the De Giorgi iteration technique to the vectorial setting. For this we replace the usual scalar truncation operator by a vectorial shortening operator. As an application, we prove local boundedness for local and nonlocal nonlinear systems. Furthermore, we show convex hull properties, which are a generalization of the maximum principle to the case of systems.

We prove an Aleksandrov–Bakelman–Pucci estimate for non-uniformly elliptic equations in non-divergence form. Moreover, we investigate the local behavior of solutions of such equations by proving local boundedness and a weak Harnack inequality. Here we impose an integrability assumption on ellipticity representing degeneracy or singularity, instead of specifying the particular structure of ellipticity.

The aim of this paper is to give the answer to the problem of characterization of acting conditions (necessary as well as sufficient) for nonlinear composition operators in some sequence spaces. We also characterize their boundedness and local boundedness. We focus on nonlinear composition operators acting to or from the space bvp(E) of all sequences of p-bounded variation; here p≥1 and E is a normed space.

Abstract The gradient of weak solutions to porous medium-type equations or systems possesses a higher integrability than the one assumed in the pure notion of a solution. We settle the critical and sub-critical, singular case and complete the program.

Abstract Fractional Rayleigh–Stokes equations can be described as the viscoelasticity of non‐Newtonian fluids behavior for a generalized second grade fluid. In this paper, we present the monotone iteration method to investigate the nonlinear fractional Rayleigh–Stokes equations from the perspective of the supersolution. More precisely, we analyze the local existence, boundedness, and convergence of nonnegative mild solutions under the superlinear growth conditions. Further, the local nonexistence results of mild solutions are also given.

A Natural Local Boundedness Result for Solutions of Elliptic-Parabolic Equations

In this paper, we first study the abstract Cauchy problem for evolution equations in a complete random normed module, and further some basic results of the abstract Cauchy problem derived from an almost surely locally bounded C0–semigroup and its infinitesimal generator are presented. Meantime, the example also exhibits the necessity of the almost surely local boundedness for such a C0–semigroup. Then, based on the above work, a characterization for an evolution equation in a complete random normed module to be well-posed is established, which improves and generalizes some known results.

The objective of this paper is to investigate a class of initial boundary value problems for inverse variational inequalities that arise from financial matters. By utilizing the energy inequality on a localized cylindrical region and the Caffarelli–Kohn–Nirenberge inequality, we establish the local boundedness and the Harnack inequality of weak solutions to the variational inequality.

Local boundedness for vectorial minimizers of non-uniform variational integrals

Local boundedness of weak solutions to an inclusion problem in Musielak–Orlicz–Sobolev spaces

The main goal of this note is to characterize the necessary and sufficient conditions for a composition operator to act between spaces of mappings of bounded Wiener variation in a normed-valued setting. The necessary and sufficient conditions for local boundedness of such operators are also discussed.

On the Local Boundedness of Solutions to the Equation −Δu + a∂zu = 0

This paper is concerned with the gradient continuity for the parabolic (1,p)-Laplace equation. In the supercritical case 2nn+2<p<∞, where n≥2 denotes the space dimension, this gradient regularity result has been proved recently by the author. In this paper, we would like to prove that the same regularity holds even for the subcritical case 1<p≤2nn+2 with n≥3, on the condition that a weak solution admits the Ls-integrability with s>n(2-p)p\\frac{n(2-p)}{p}$$\\end{document}]]>. The gradient continuity is proved, similarly to the supercritical case, once the local gradient bounds of solutions are verified. Hence, this paper mainly aims to show the local boundedness of a solution and its gradient by Moser’s iteration. The proof is completed by considering a parabolic approximate problem, verifying a comparison principle, and showing a priori gradient estimates of a bounded weak solution to the relaxed equation.

Abstract This paper deals with regularity properties for variational integrals with the splitting structure of the form where , is the adjugate matrix of order , and , , , are Carathéodory functions satisfying suitable structural conditions. Local integrability, local boundedness, and local Hölder continuity for local minimizers are derived.

Abstract We consider the mixed local and nonlocal functionals with nonstandard growth with , , and being a bounded domain. We study, utilizing expansion of positivity, local behavior of the minimizers of such problems, involving local boundedness, local Hölder continuity, and Harnack inequality. This solves a problem raised by De Filippis and Mingione [Math. Ann. 388 (2024) 262–328].

Local Boundedness for Minimizers of Anisotropic Functionals with Monomial Weights