Related papers: The Time Derivative in a Singular Parabolic Equati…

We establish the boundedness of time derivatives of solutions to parabolic $p$-Laplace equations. Our approach relies on the Bernstein technique combined with a suitable approximation method. As a consequence, we obtain an optimal…

We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on…

We study quasilinear evolutionary partial integro-differential equations of second order which include time fractional $p$-Laplace equations of time order less than one. By means of suitable energy estimates and De Giorgi's iteration…

The aim of this paper is to study time-fractional pseudo-parabolic type equations generated by the Dunkl operator. The forward problem is considered and its well-posedness is established. In particular, a prior estimates are obtained in the…

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by L\'evy processes of the type $$ \partial^{\alpha}_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}}…

In this paper, we study parabolic equations in divergence form with coefficients that are singular degenerate as some Muckenhoupt weight functions in one spatial variable. Under certain conditions, weighted reverse H\"{o}lder's inequalities…

This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace…

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1<p<2$ and fractional exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for data in the…

This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…

We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and H\"older continuity of weak solutions to this…

We derive weighted Sobolev-Poincar\'e type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class,…

We consider the evolutionary symmetric $p$-Laplacian with safety $1$. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior…

We study the discrete version of the $p$-Laplacian. Based on its variational properties we discuss some features of the associated parabolic problem. Our approach allows us in turn to obtain interesting information about positivity and…

We first prove the equivalence of two definitions of Riemann-Liouville fractional integral on time scales, then by the concept of fractional derivative of Riemann-Liouville on time scales, we introduce fractional Sobolev spaces,…

Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.

We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and…

We consider the evolutionary $p$-Laplace equation in $\mathbb{R}^n$. For $p>n$, we construct a solution $u$ with a moving gradient singularity in the sense that $|\nabla u(x,t)|\to \infty$ for each $t$ as $x\to\xi(t)$, where…

In this paper we answer Iwaniec and Sbordone's conjecture \cite{IB94} concerning very weak solutions to the $p$-Laplace equation. Namely, on one hand we show that distributional solutions of the $p$-Laplace equation in $W^{1,r}$ for $p \neq…

We study the asymptotics of the Schr\"odinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a…

In the present paper we consider an inverse source problem for time-fractional mixed parabolic-hyperbolic equation with the Caputo derivative. In case, when hyperbolic part of the considered mixed type equation is wave equation, the…