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We study the well-posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions. We give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3D under null condition, with small scaling invariant $dot{W}^{2,1}(mathbb{R}^3)$ data. We obtain a bilinear estimate that is effective to the hyperbolic systems which do not have any time decay. It allows us to achieve the boundedness of the weighted BV norm of the radial solution.
The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^infty (mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform c
This paper deals with the existence of positive solutions for the nonlinear system q(t)phi(p(t)u_{i}(t)))+f^{i}(t,textbf{u})=0,quad 0<t<1,quad i=1,2,...,n. This system often arises in the study of positive radial solutions of nonlinear elliptic syste
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
In this paper we prove the almost sure existence of global weak solution to the 3D incompressible Navier-Stokes Equation for a set of large data in $dot{H}^{-alpha}(mathbb{R}^{3})$ or $dot{H}^{-alpha}(mathbb{T}^{3})$ with $0<alphaleq 1/2$. This is ac
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