Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R^3
174
0
0.0
(
0
)
Authors
Bixiang Wang
Ask ChatGPT about the research
No Arabic abstract
The existence of a random attractor in H^1(R^3) times L^2(R^3) is proved for the damped semilinear stochastic wave equation defined on the entire space R^3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method.
rate research
Read More
This paper is concerned with the nonlinear damped wave equation on a measure space with a self-adjoint operator, instead of the standard Laplace operator. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear estimates which generalize the so-called Matsumura estimates, and prove the small data global existence of solutions to the damped wave equation based on the linear estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrodinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.
This paper is devoted to the study of asymptotic behaviors of solutions to the one-dimensional defocusing semilinear wave equation. We prove that finite energy solution tends to zero in the pointwise sense, hence improving the averaged decay of Lindblad and Tao. Moreover, for sufficiently localized data belonging to some weighted energy space, the solution decays in time with an inverse polynomial rate. This confirms a conjecture raised in the mentioned work. The results are based on new weighted vector fields as multipliers applied to regions enclosed by light rays. The key observation for the first result is an integrated local energy decay for the potential energy, while the second result relies on a type of weighted Gagliardo-Nirenberg inequality.
In this paper, we use Dafermos-Rodnianskis new vector field method to study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in $mathbb{R}^{1+3}$. We prove that the solution decays as quickly as linear waves for $p>frac{1+sqrt{17}}{2}$, covering part of the sub-conformal case, while for the range $2<pleq frac{1+sqrt{17}}{2}$, the solution still decays with rate at least $t^{-frac{1}{3}}$. As a consequence, the solution scatters in energy space when $p>2.3542$. We also show that the solution is uniformly bounded when $p>frac{3}{2}$.
This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities begin{equation*} -left(a+bint_{mathbb{R}^{3}}| abla u(x)|^{2}right) Delta u =lambda u+|u|^{p-2}u+u^{5}quad text{for some} lambdainmathbb{R},quad xinmathbb{R}^{3}, end{equation*} with prescribed $L^{2}$-norm mass begin{equation*} int_{mathbb{R}^{3}}u^{2}=c^{2} end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_{c},lambda_{c})in S(c)times mathbb{R}$ for any $c>0$, $a,b >0$ and $frac{14}{3}leq p< 6,$ where $S(c)={uin H^{1}(mathbb{R}^{3}):int_{mathbb{R}^{3}}u^{2}=c^{2}}.$ textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states oindent{AMS Subject Classification:, 37L05; 35B40; 35B41.}
This thesis is concerned with the asymptotic behavior of solutions of stochastic $p$-Laplace equations driven by non-autonomous forcing on $mathbb{R}^n$. Two cases are studied, with additive and multiplicative noise respectively. Estimates on the tails of solutions are used to overcome the non-compactness of Sobolev embeddings on unbounded domains, and prove asymptotic compactness of solution operators in $L^2(mathbb{R}^n)$. Using this result we prove the existence and uniqueness of random attractors in each case. Additionally, we show the upper semicontinuity of the attractor for the multiplicative noise case as the intensity of the noise approaches zero.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
