Related papers: Asymptotic behavior for the critical nonhomogeneou…
We study the large time behavior of solutions to the porous medium equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u^m, \quad \hbox{in} \ \real^N\times(0,\infty), $$ where $m>1$ and $N\geq3$.…
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 $\sigma>0$…
We study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u, \quad \hbox{in} \ \real^N\times(0,\infty). $$ The asymptotic behavior proves to…
We study a nonlinear porous medium type equation involving the infinity Laplacian operator. We first consider the problem posed on a bounded domain and prove existence of maximal nonnegative viscosity solutions. Uniqueness is obtained for…
In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} \mu(|u|)$, where $\mu$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020))…
We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption $$ \partial_{t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ with…
We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma >…
We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having nonhomogeneous in space and time source terms with zero mean. In dimensions two and three, we…
Kamin and V\'azquez proved in 1991 that solutions to the Cauchy-Dirichlet problem for the porous medium equation $u_t=(u^m)_{xx}$ on the half line with zero boundary data and nonnegative compactly supported integrable initial data behave…
In this paper we obtain higher order asymptotic profilles of solutions to the Cauchy problem of the linear damped wave equation in $\textbf{R}^n$ \begin{equation*} u_{tt}-\Delta u+u_t=0, \qquad u(0,x)=u_0(x), \quad u_t(0,x)=u_1(x),…
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is…
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by…
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…
In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u =…
We consider the large time behavior of the solutions to the Cauchy problem for the BBM-Burgers equation. We prove that the solution to this problem goes to the self-similar solution to the Burgers equation called the nonlinear diffusion…
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case…
Consider the anisotropic porous medium equation, $u_t=\sum\limits_{i=1}^n(u^{m_i})_{x_ix_i},$ where $m_i>0, (i=1,2,...,n)$ satisfying $\min\limits_{1\le i\le n}\{m_i\}\le 1,$ $\sum\limits_{i=1}^nm_i>n-2,$ and $\max\limits_{1\le i\le…
We study existence and uniqueness of bounded solutions to a fractional nonlinear porous medium equation with a variable density, in one space dimension.
We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the…
In this paper, we obtain several asymptotic profiles of solutions to the Cauchy problem for structurally damped wave equations $\partial_{t}^{2} u - \Delta u + \nu (-\Delta)^{\sigma} \partial_{t} u=0$, where $\nu >0$ and $0< \sigma \le1$.…