Related papers: An inhomogeneous nonlinear parabolic equation with…

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

Consider the heat equation $u_t-\Delta u=0$ on a bounded $C^2$ domain $\Omega$ in $\mathbb{R}^{n}(n\geq 2)$ with any positive initial data. If a superlinear radiation law $\frac{\partial u}{\partial n}=u^{q}$ with $q>1$ is imposed on a…

We present a complete description of the similarity solutions $u_{\alpha}(x,t)=t^{-\alpha/2}f(\Vert x \Vert/\sqrt{t};\alpha)$ for the following nonlinear diffusion equation $$ u_{t}+\gamma\vert u_{t} \vert =\Delta u\qquad(-1<\gamma<1) $$…

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of…

We study the following nonlinear scalar field equation $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and…

We produce a finite time blow-up solution for nonlinear fractional heat equation ($\partial_t u + (-\Delta)^{\beta/2}u=u^k$) in modulation and Fourier amalgam spaces on the torus $\mathbb T^d$ and the Euclidean space $\mathbb R^d.$ This…

Let $(M,g(t))$ be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type $$\ppt f=\Delta f-f \ln f +Rf.$$…

In this paper, we consider the semilinear heat equations under Dirichlet boundary condition \[ u_{t}\left(x,t\right)=\Delta u\left(x,t\right)+f(u(x,t)), & \left(x,t\right)\in \Omega\times\left(0,+\infty\right), u\left(x,t\right)=0, &…

We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial…

We consider the Dirichlet problem for the energy-critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^5,~&\mbox{ in } \Omega \times \mathbb{R}^+,\\ u(x,t)=0,~&\mbox{ on } \partial \Omega \times \mathbb{R}^+,\\…

Long time dynamics of solutions to the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ on $\R^6\times(0,\infty)$ is investigated. It is shown that there exists a radially symmetric global solution $u(x,t)\in C([0,\infty);\dot…

This paper is devoted to the analysis of the following nonlinear wave equation \[ u_{tt} - u_{xx} + (1 + q\delta_0(x)) \sin u = 0, \] where $\delta_0 = \delta_0(x)$ is the Dirac delta function centered at the origin and $q \in \mathbb{R}$…

This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=\Delta (\gamma (v) u ) &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] v_t=\Delta v - v + u &\mathrm{in}\ \Omega\times(0,\infty),…

In this paper, we consider the semilinear damped wave equation with nonlinearities of derivative type $|u_t|^p$. We observe that this problem admits a unique global (in time) solution with small initial data for all $p > 1$ in low spatial…

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,…

In this paper we prove that positive weak solutions for quasilinear parabolic equations on bounded domains subject to homogenous Neumann boundary conditions becme classical and global under the unique condition that the reaction doesn't…

In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ \Delta^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data…

We consider solutions $u(t)$ to the 3d NLS equation $i\partial_t u + \Delta u + |u|^2u=0$ such that $\|xu(t)\|_{L^2} = \infty$ and $u(t)$ is nonradial. Denoting by $M[u]$ and $E[u]$, the mass and energy, respectively, of a solution $u$, and…