Related papers: Reduced measures associated to parabolic problems

Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2\Delta v+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\…

In this paper we are concerned with a general singular Dirichlet boundary value problem whose model is the following $$ \begin{cases} -\Delta u = \frac{\mu}{u^{\gamma}} & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0 &\text{on}\…

We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and…

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity…

In this paper we prove the existence of multiple nontrivial solutions of the following equation. \begin{align*} \begin{split} -\Delta_{p}u & = \lambda |u|^{q-2}u+f(x,u)+\mu\,\,\mbox{in}\,\,\Omega, u & = 0\,\, \mbox{on}\,\, \partial\Omega;…

This article studies the partial H\"older continuity of weak solutions to certain degenerate parabolic systems whose model is the differentiable parabolic $p(x,t)$-Laplacian system, \begin{equation*}\partial_t…

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-\Delta)^s v = \mu \quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure…

The paper concentrates on the application of the following Hardy inequality \begin{equation*} \int_\Omega \ |\xi(x)|^p \omega_{1 }(x)dx\le \int_\Omega |\nabla \xi(x)|^p\omega_{2 }(x)dx, \end{equation*} to the proof of existence of weak…

In this article, we will prove existence results for the equations of the type $-\Delta_{N}u=H_{l}(u)+\mu$ and $F_{\frac{N}{2}}[-u]=H_{l}(u)+\mu$ in a bounded domain $\Omega$, with Dirichlet boundary condition, where the source term…

For parabolic equations of the form $$ \frac{\partial u}{\partial t} - \sum_{i,j=1}^n a_{ij} (x, u) \frac{\partial^2 u}{\partial x_i \partial x_j} + f (x, u, D u) = 0 \quad \mbox{in } {\mathbb R}_+^{n+1}, $$ where ${\mathbb R}_+^{n+1} =…

The purpose of this work is to analyze the well-posedness and blow-up behavior of solutions to the nonlocal semilinear parabolic equation with a forcing term: \[ \partial_t u - \Delta u = \|u(t)\|_{q}^\alpha |u|^p + t^{\varrho}…

We investigate the local boundedness of solutions $u:\Omega_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*} \partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }\Omega_T=\Omega\times(0,T) \end{equation*} that…

The aim of the paper is to study the problem $u_{tt}-c^2\Delta u=0$ in $\mathbb{R}\times\Omega$, $\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa v+\rho u_t =0$ on $\mathbb{R}\times \Gamma_1$, $v_t =\partial_\nu u$…

For the parabolic obstacle-problem-like equation $$\Delta u - \partial_t u = \lambda_+ \chi_{\{u>0\}} - \lambda_- \chi_{\{u<0\}} ,$$ where $\lambda_+$ and $\lambda_-$ are positive Lipschitz functions, we prove in arbitrary finite dimension…

In this paper the existence of solutions, $(\lambda,u)$, of the problem $$-\Delta u=\lambda u -a(x)|u|^{p-1}u \quad \hbox{in }\Omega, \qquad u=0 \quad \hbox{on}\;\;\partial\Omega,$$ is explored for $0 < p < 1$. When $p>1$, it is known that…

This article studies the continuity of bounded nonnegative weak solutions to inhomogeneous doubly nonlinear parabolic equations. A model equation is \begin{equation*}\partial_t u-\operatorname{div}(u^{m-1}|Du|^{p-2}Du)=f\qquad…

We establish a local null controllability result for following the nonlinear parabolic equation: $$u_t-\left(b\left(x,\int_0^1u \ \right)u_x \right)_x+f(t,x,u)=h\chi_\omega,\ (t,x)\in (0,T)\times (0,1) $$ where $b(x,r)=\ell(r)a(x)$ is a…

For the Schr\"odinger equation, $ (i \partial_t + \Delta) u = 0 $ on a torus, an arbitrary non-empty open set $ \Omega $ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T]…

We show that the parabolic equation $u_t + (-\Delta)^s u = q(x) |u|^{\alpha-1} u$ posed in a time-space cylinder $(0,T) \times \mathbb{R}^N$ and coupled with zero initial condition and zero nonlocal Dirichlet condition in $(0,T) \times…