Related papers: Bounded solutions and interpolative gap bounds for…

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 nonlinear elliptic equations modeled on \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu, \] where $2\le p<q<\infty$, $a(\cdot) \ge 0$, and $\mu$ is a signed Borel measure with finite total mass. We prove local…

We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by $H(x,t)=t^p+a(x)t^q$ with $1<p\leq q$ and $a(x)$ a nonnegative $C^{0,\alpha}$-continuous function. Our main result…

We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_{2}$ weight. We obtain representations and boundary traces for solutions…

We consider a mixed type boundary value problem for a class of degenerate parabolic-hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary…

We establish the local Lipschitz regularity in space for the viscosity solutions to the parabolic double phase equation of the form \[ \smash{\partial_{t}u-\operatorname{div} \left(|Du|^{p-2}D u+a(z)|D u|^{q-2}D u\right)=f(z, Du)} \] by…

We study a series of regularity results for solutions to a degenerate or singular fully nonlinear integro-differential equation of the form $$- \big( \sigma_{1}(|Du|) + a(x) \sigma_{2}(|Du|) \big) \mathcal{I}_{\tau}(u,x) = f(x).$$ In the…

In this paper, we consider the asymptotic behavior of traveling wave solutions of the degenerate nonlinear parabolic equation: $u_{t}=u^{p}(u_{xx}+u)-\delta u$ ($\delta = 0$ or $1$) for $\xi \equiv x - ct \to - \infty$ with $c>0$. We give a…

In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_{\Omega} \left( |D^m…

The aim of this paper is to establish a higher integrability result for very weak solutions of certain parabolic systems whose model is the parabolic $p(x,t)$-Laplacian system. Under assumptions on the exponent function…

Let $\Omega$ be a bounded domain in ${\mathbb R}^N$ and $T>0$. We study the problem \begin{equation} (P)\left\{ \begin{array}{lll} u_t - \Delta u \pm g(u) &= \mu \quad &\text{in } Q_T:=\Omega \times (0,T) \\ \phantom{------,} u&=0 &\text{on…

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 establish the local H\"older regularity of the spatial gradient of bounded weak solutions $u\colon E_T\to\R^k$ to the non-linear system of parabolic type \begin{equation*} \partial_tu-\Div\Big(…

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$ -{\rm div}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty, a(x)\ge0. $$ We find some…

In this study, we firstly establish the well-posedness of a degenerate parabolic equation under Dirichlet boundary conditions. Following this, we introduce a shape design problem, which acts as a framework for approximating the degenerate…

We consider a class of degenerate elliptic fully nonlinear equations with applications to Grad equations: \begin{align} \begin{cases} |Du|^\gamma \mathcal{M}_{\lambda,\Lambda}^+\big(D^2u(x)\big)=f\big(|u\geq u(x)|\big) &\text{ in }\Omega,…

We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions…

We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularity $$ \partial_t u=[|D u|^q+a(x,t)|D u|^s]\left(\Delta u+(p-2)\left\langle D^2 u\frac{D u}{|D u|},\frac{D u}{|D…

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity $$u_t-k\Delta u_t-\Delta u=I^\gamma_{0+}(|u|^{p})+\omega(x),\,\ (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ where $p>1,\,k\geq 0$,…

The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times…