Related papers: Stability of solutions of quasilinear parabolic eq…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We first study the problem \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\…

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega \times(0,T).$ We study problems of the model type \[ \left\{ \begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u=\mu\qquad\text{in }Q,\\ {u}=0\qquad\text{on…

We study continuous dependence of solutions to quasilinear evolution equations of parabolic-type in the framework of maximal $L^p$-regularity. For equations of the form \[ \frac{d\phi}{dt} + A(t,\phi)\phi = f(t,\phi), \] we establish…

We investigate the following quasilinear parabolic and singular equation, {equation} \tag{{\rm P$_t$}} \{{aligned} & u_t-\Delta_p u =\frac{1}{u^\delta}+f(x,u)\;\text{in}\,(0,T)\times\Omega, & u =0\,\text{on}…

We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ We prove that if a solution $u$ of this equation is bounded and its initial value $u(x,0)$ has…

This is a continuation, and conclusion, of our study of bounded solutions $u$ of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line whose initial data $u_0=u(\cdot,0)$ have finite limits $\theta^\pm$ as $x\to\pm\infty$. We…

The current paper considers the boundedness of solutions to the following quasilinear Keller-Segel model (with logistic source) $$\left\{\begin{array}{ll} u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in…

We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution…

In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{\beta-1}u+\alpha|u|^{q-2}u, \] where $p>1,\beta>0$, $q\geq1$ and $\alpha>0$.…

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…

We investigate quasilinear discrete PDEs $\partial_t u = \Delta^N \varphi(u)+ Kf(u)$ of reaction-diffusion type with nonlinear diffusion term defined on an $n$-dimensional unit torus discretized with mesh size $\tfrac1N$ for $N\in {\mathbb…

In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this…

We consider a parabolic equation of the form u_t=\Delta u +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions,…

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-\Delta _{p(x)}u = f(x,u)$ in $ (0,T)\times\Omega$; $u = 0$ on $(0,T)\times\partial\Omega$; $u(0,x)=u_0(x)$ in $\Omega$;…

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t…

We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and…

The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum…

We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…

In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but,…

This paper deals with the asymptotic behavior as $t\rightarrow T<\infty$ of all weak (energy) solutions of a class of equations with the following model representative: \begin{equation*} (|u|^{p-1}u)_t-\Delta_p(u)+b(t,x)|u|^{\lambda-1}u=0…