Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal existence for some 4-D quasilinear wave equations with low regularity
135
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in $H^{3}times H^{2}$. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.
rate research
Read More
Hormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $Box u = Q(u, u, u)$ where $Q$ vanishes to second order and $(partial_u^2 Q)(0,0,0)=0$. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms $upartial_alpha u = frac{1}{2}partial_alpha u^2$ and as such do not immediately generalize to systems. The current study remedies such and extends both results to the case of multiple speed systems.
We study the global existence of solutions to semilinear damped wave equations in the scattering case with derivative power-type nonlinearity on (1+3) dimensional nontrapping asymptotically Euclidean manifolds. The main idea is to exploit local energy estimate, together with local existence to convert the parameter $mu$ to small one.
We study the asymptotic behavior of solutions to wave equations with a structural damping term [ u_{tt}-Delta u+Delta^2 u_t=0, qquad u(0,x)=u_0(x), ,,, u_t(0,x)=u_1(x), ] in the whole space. New thresholds are reported in this paper that indicate which of the diffusion wave property and the non-diffusive structure dominates in low regularity cases. We develop to that end the previous authors research in 2019 where they have proposed a threshold that expresses whether the parabolic-like property or the wave-like property strongly appears in the solution to some regularity-loss type dissipative wave equation.
We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove almost global existence with sharp lower bound of the lifespan for the four dimensional critical problem.
In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
