Related papers: Renormalized Solution for the Nonlinear Parabolic …

In this work we study the following nonlocal problem \begin{equation*} \left\{ \begin{aligned} M(\|u\|^2_X)(-\Delta)^s u&= \lambda {f(x)}|u|^{\gamma-2}u+{g(x)}|u|^{p-2}u &&\mbox{in}\ \ \Omega, u&=0 &&\mbox{on}\ \ \mathbb R^N\setminus…

The main purpose of this article is to reconstruct the nonnegative coefficient $a$ in the double phase problem $\mathrm{div}\,(|\nabla u|^{p-2}\nabla u+a|\nabla u|^{q-2}\nabla u)=0$ in a domain $\Omega$, $u=f$ on $\partial\Omega$, from the…

It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+|\nabla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ in $\mathbb{R}^n_+\times[0,T]$ and $u(x,0)=0$ in $\mathbb{R}^n_+$ under certain conditions…

In this article, we consider the following problem: $$ \quad \left\{ \begin{array}{lr} \quad (-\Delta)^s u = \alpha u^+ -\beta u^{-} + f(u) + h \; \text{in}\;\Omega \quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus \Omega,…

In this work, we establish the existence and multiplicity of weak solutions for nonlocal elliptic problems driven by the fractional $\Phi$-Laplacian operator, in the presence of a sign-indefinite nonlinearity. More specifically, we…

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for…

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown…

This paper deals with the existence and multiplicity of solutions for the generalized $(p, q)$-Laplacian equation \begin{align*} &-{\text{ div}}(A(x, u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x, u)|\nabla u|^p -{\text{ div}}(B(x, u)|\nabla…

In this paper we deal with local estimates for parabolic problems in $\mathbb{R}^N$ with absorbing first order terms, whose model is $\{ {l} u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{in}\, (0,T) \times \mathbb{R}^N\,, \\[1.5 ex]…

A time-fractional Fokker-Planck initial-boundary value problem is considered, with differential operator $u_t-\nabla\cdot(\partial_t^{1-\alpha}\kappa_\alpha\nabla u-\textbf{F}\partial_t^{1-\alpha}u)$, where $0<\alpha <1$. The forcing…

This paper examines the behavior of a positive solution $u\in C^{1,\alpha}(\Bar{\Omega})$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation: \begin{equation*}…

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -\psi \left [ \int_\Omega \left ( \frac{|\nabla u |^p}{p} + \mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) =…

We study the long-time asymptotics of solutions of the uniformly parabolic equation \[ u_t + F(D^2u) = 0 \quad {in} \R^n\times \R_+, \] for a positively homogeneous operator $F$, subject to the initial condition $u(x,0) = g(x)$, under the…

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_{\Omega}|\nabla u|^n)\Delta_n u = f(x,u) \; \text{in}\; \Omega,\quad u…

The purpose of the article is to study the existence, regularity, stabilization and blow up results of weak solution to the following parabolic $(p,q)$-singular equation: \begin{equation*} (P_t)\; \left\{\begin{array}{rllll} u_t-\Delta_{p}u…

We will prove several existence and regularity results for the mixed local-nonlocal parabolic equation of the form \begin{eqnarray} \begin{split} u_t-\Delta u+(-\Delta)^s u&=\frac{f(x,t)}{u^{\gamma(x,t)}} \text { in } \Omega_T:=\Omega…

The aim of this paper is investigating the existence of weak bounded solutions of the gradient-type quasilinear elliptic system $$(P)\qquad \left\{ \begin{array}{ll} - {\rm div} ( a_i(x, u_i, \nabla u_i) ) + A_{i, t} (x, u_i, \nabla u_i) =…

We obtain new a priori estimates for the nonnegative solutions of the equation \[ u_{t}-\Delta u+|\nabla u|^{q}=0 \] in $Q_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $\Omega=\mathbb{R}^{N},$ or $\Omega$ is…

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda…

In this paper, we are interested in the regularity of weak solutions $u\colon\Omega_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $\Omega_T$},…