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Register a new userProfiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrodinger equations
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Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrodinger equation.
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In this paper, we consider the following nonlinear Schr{o}dinger equations with mixed nonlinearities: begin{eqnarray*} left{aligned &-Delta u=lambda u+mu |u|^{q-2}u+|u|^{2^*-2}uquadtext{in }mathbb{R}^N, &uin H^1(bbr^N),quadint_{bbr^N}u^2=a^2, endalignedright. end{eqnarray*} where $Ngeq3$, $mu>0$, $lambdainmathbb{R}$ and $2<q<2^*$. We prove in this paper begin{enumerate} item[$(1)$]quad Existence of solutions of mountain-pass type for $N=3$ and $2<q<2+frac{4}{N} $. item[$(2)$]quad Existence and nonexistence of ground states for $2+frac{4}{N}leq q<2^*$ with $mu>0$ large. item[$(3)$]quad Precisely asymptotic behaviors of ground states and mountain-pass solutions as $muto0$ and $mu$ goes to its upper bound. end{enumerate} Our studies answer some questions proposed by Soave in cite[Remarks~1.1, 1.2 and 8.1]{S20}.
In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schrodinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
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), end{equation*} where $nintextbf{N}$ and $u_0$, $u_1in L^2(textbf{R}^n)$. Established hyperbolic part of asymptotic expansion seems to be new in the sense that the order of the expansion of the hyperbolic part depends on the spatial dimension.
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 the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
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