Related papers: Asymptotic Behavior for a Nonlocal Diffusion Equat…
We are considering the asimptotic behavior as $t\to\infty$ of solutions of the Cauchy problem for parabolic second order equations with time periodic coefficients. The problem is reduced to considering degenerate time-homogeneous diffusion…
We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We…
The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a…
In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding…
In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates…
This article discusses the analyticity and the long-time asymptotic behavior of solutions to space-time fractional diffusion equations in $\mathbb{R}^d$. By a Laplace transform argument, we prove that the decay rate of the solution as…
The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption $$\partial_t u-\Delta u^m+|x|^{\sigma}u^q=0, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N,$$ with $m\geq1$,…
Consider the Cauchy problem for a nonlinear diffusion equation \begin{equation} \tag{P} \left\{ \begin{array}{ll} \partial_t u=\Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0)=\lambda+\varphi(x)>0 &…
Ultrasonic propagation through media with thermal and molecular relaxation can be modeled by third-order in time nonlinear wave-like equations with memory. This paper investigates the asymptotic behavior of a Cauchy problem for such a…
This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat…
In this work, we consider a nonlocal Fisher-KPP reaction-diffusion problem with Neumann boundary condition and nonnegative initial data in a bounded domain in $\mathbb{R}^n (n \ge 1)$, with reaction term $u^\alpha(1-m(t))$, where $m(t)$ is…
We examine the long-term asymptotic behavior of dissipating solutions to aggregation equations and Patlak-Keller-Segel models with degenerate power-law and linear diffusion. The purpose of this work is to identify when solutions decay to…
We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $\sigma>0$…
In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is…
We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of…
We study the large-time behavior in all $L^p$ norms of solutions to an inhomogeneous nonlocal heat equation in $\mathbb{R}^N$ involving a Caputo $\alpha$-time derivative and a power $\beta$ of the Laplacian when the dimension is large, $N>…
The long-time asymptotics of solutions of the Cauchy problem for the heat equation are constructed in the case when the initial function at infinity has power asymptotics.
The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem $\partial_t u = \Delta_p u + |\nabla u|^q$ when the initial data converge to zero at infinity. Sufficient conditions on…
In this paper, we study the time-space fractional differential equation of the Volterra type: \begin{align*} {D}^\alpha_{0 \vert t} (u) +(-\Delta_N)^{\sigma}u &= u(1+au-bu^2)-au\int_0^t {K}(t-s) u(\cdot) \, ds, \end{align*} where $a,b>0$…
In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are…