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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.
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.
We consider the two-dimensional quasilinear wave equations with standard null-form type quadratic nonlinearities. We prove global wellposedness without using the Lorentz boost vector fields.
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 e
We get a local existence result in $H^s$ with $s>3/2$ for second order quasilinear wave equation with radial initial data in 2+1 dimensions, based on an improvement of Strichartz estimate in the radial case. Moreover, we get the corresponding local w
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 energ