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A doubly degenerate parabolic equation in non-divergent form with variable growth is investigated in this paper. In suitable spaces, we prove the existence of weak solutions of the equation for cases $1\leq m < 2$ and $m\geq 2$ in different…

We study whether the solutions of a parabolic equation with diffusion given by the fractional Laplacian and a dominating gradient term satisfy Dirichlet boundary data in the classical sense or in the generalized sense of viscosity…

A boundary value problem related to a third- order parabolic equation with a small parameter is analized. This equation models the one-dimensional evolution of many dissipative media as viscoelastic fluids or solids, viscous gases,…

We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on…

We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time- dependent. Moreover, we study the asymptotic behaviour…

The present paper is concerned with the Cauchy-Dirichlet problem for fractional (and non-fractional) nonlinear diffusion equations posed in bounded domains. Main results consist of well-posedness in an energy class with no sign restriction…

Existence and uniqueness of solutions is shown for a class of viscoelastic flows in porous media with particular attention to problems with nonsmooth porosities. The considered models are formulated in terms of the time-dependent nonlinear…

This article presents a comprehensive overview and supplement to recent developments in second-order elliptic partial differential equations formulated in double divergence form, along with an exploration of their parabolic counterparts.

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which…

We study the closure of approximating sequences of some diffusion equations under certain weak convergence. A specific description of the closure under weak $H^1$-convergence is given, which reduces to the original equation when the…

Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for…

This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…

We prove a Carleman estimate for a one-dimensional parabolic equation which degenerates at one extremity of the domain and has a bounded, time dependent coefficient multiplying the diffusion term. Then we use the estimate to show the null…

A class of linear degenerate elliptic equations inspired by nonlinear diffusions of image processing is considered. It is characterized by an interior degeneration of the diffusion coefficient. It is shown that no particularly natural,…

A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly…

We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…

When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be…

This note gives a first sufficient condition that insures a non-negative, locally bounded, local solution to a logarithmically singular parabolic equation is continuous at a vanishing point and an estimate of the modulus of continuity is…

This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case. where the drift is a…

We provide pointwise upper bounds for the transition kernels of semigroups associated with a class of systems of nondegenerate elliptic partial differential equations with unbounded coefficients with possibly unbounded diffusion…