Global existence and decay estimate of solution to compressible quantum Navier-Stokes equations with damping

In this paper, we consider the Cauchy problem of the compressible quantum Navier-Stokes equations with damping in R3. We first assume that the H3-norm of the initial data is sufficiently small while the higher derivative can be arbitrarily large, and prove the global existence of smooth solutions. Then the decay estimate of the solution is derived for the initial data in a homogeneous Sobolev space or Besov space with negative exponent. In addition, the usual Lp−L2(1 ≤ p ≤ 2) type decay rate is obtained without assuming that the Lpnorm of the initial data is sufficiently small.

Existence and asymptotic behavior of Radon measure-valued solutions for a class of nonlinear parabolic equations

In this paper we address the weak Radon measure-valued solutions associated with the Young measure for a class of nonlinear parabolic equations with initial data as a bounded Radon measure. This problem is described as follows: $$ \textstyle\begin{cases} u_{t}=\alpha u_{xx}+\beta [\varphi (u) ]_{xx}+f(u) &\text{in} \ Q:=\Omega \times (0,T), \\ u=0 &\text{on} \ \partial \Omega \times (0,T), \\ u(x,0)=u_{0}(x) &\text{in} \ \Omega , \end{cases} $$ { u t = α u x x + β [ φ ( u ) ] x x + f ( u ) in Q : = Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where $T>0$ T > 0 , $\Omega \subset \mathbb{R}$ Ω ⊂ R is a bounded interval, $u_{0}$ u 0 is nonnegative bounded Radon measure on Ω, and $\alpha , \beta \geq 0$ α , β ≥ 0 , under suitable assumptions on φ and f. In this work we prove the existence and the decay estimate of suitably defined Radon measure-valued solutions for the problem mentioned above. In particular, we study the asymptotic behavior of these Radon measure-valued solutions.

Strichartz estimates for the Schrödinger equation with a measure-valued potential

We prove Strichartz estimates for the Schrödinger equation in R n \mathbb {R}^n , n ≥ 3 n\geq 3 , with a Hamiltonian H = − Δ + μ H = -\Delta + \mu . The perturbation μ \mu is a compactly supported measure in R n \mathbb {R}^n with dimension α > n − ( 1 + 1 n − 1 ) \alpha > n-(1+\frac {1}{n-1}) . The main intermediate step is a local decay estimate in L 2 ( μ ) L^2(\mu ) for both the free and perturbed Schrödinger evolution.

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

Global existence and asymptotic behavior of solutions to fractional ( p , q )-Laplacian equations

The aim of this paper is to consider the following fractional parabolic problem u t + ( − Δ ) p α u + ( − Δ ) q β u = f ( x , u ) ( x , t ) ∈ Ω × ( 0 , ∞ ) , u = 0 ( x , t ) ∈ ( R N ∖ Ω ) × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) x ∈ Ω , where Ω ⊂ R N is a bounded domain with Lipschitz boundary, ( − Δ ) p α is the fractional p-Laplacian with 0 < α < 1 < p < ∞, ( − Δ ) q β is the fractional q-Laplacian with 0 < β < α < 1 < q < p < ∞, r > 1 and λ > 0. The global existence of nonnegative solutions is obtained by combining the Galerkin approximations with the potential well theory. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions.

Long-time behavior of solutions to a fourth-order nonlinear Schrödinger equation with critical nonlinearity

We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.