Related papers: Extinction for a singular diffusion equation with …
Consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p+h(x)\ \ \text{ in } \Omega\times(0,T)$$ with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question…
This paper is devoted to studying the local behavior of non-negative weak solutions to the doubly non-linear parabolic equation \begin{equation*} \partial_t u^q - \text{div}\big(|D u|^{p-2}D u\big) = 0 \end{equation*} in a space-time…
We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…
We consider the asymptotic behaviour of finite energy solutions to the one-dimensional defocusing nonlinear wave equation $-u_{tt} + u_{xx} = |u|^{p-1} u$, where $p > 1$. Standard energy methods guarantee global existence, but do not…
In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…
We are concerned with singular elliptic equations of the form $-\Delta u= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with…
We consider a class of porous medium type of equations with Caputo time derivative. The prototype problem reads as $\Dc u=-\A u^m$ and is posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with zero Dirichlet boundary…
Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and…
We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^{\alpha}(x,\partial{}B)\Phi_p(\nabla u)) = b(x)f(u)$ in a ball B,…
The large time behaviour of nonnegative solutions to a quasilinear degenerate diffusion equation with a source term depending solely on the gradient is investigated. After a suitable rescaling of time, convergence to a unique profile is…
In this paper, we consider the defocusing nonlinear wave equation $-\partial_t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb R\times \mathbb R^d$. Building on our companion work ({\it \small Self-similar imploding solutions of the relativistic Euler…
If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta…
The asymptotic behavior, as $n\rightarrow \infty $ of the conditional distribution of the number of particles in a decomposable critical branching process $\mathbf{Z}% (m)=(Z_{1}(m),...,Z_{N}(m)),$ with $N$ types of particles at moment…
This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point…
We investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension. Depending on the branching and annihilation rates, a few-particle initial state can evolve to a propagating finite…
Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-\Delta u^m+|x|^{\sigma}u^q = 0 \;\;\text{ in }\;\;…
We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=\Delta…
Consider classical solutions to the parabolic reaction diffusion equation $$ &u_t =Lu+f(x,u), (x,t)\in R^n\times(0,\infty); &u(x,0) =g(x)\ge0, x\in R^n; &u\ge0, $$ where $$ L=\sum_{i,j=1}^na_{i,j}(x)\frac{\partial^2}{\partial x_i \partial…
We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to…
We study existence of global solutions and finite time blow-up of solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term $\rho(x) u^p$ with $p>1$; this is a…