Some Study On Damped Wave Equations

Firstly,in this paper we focus on the damped semilinear wave equations oU_tside the unit ball with Neumann boundary condition U_tt-?u + U_t =|u|p.Blow up result is established assuming that the power p of the nonlinear term |u|p satisfes:1<p ? pN*= 1 + 2/N,which is known as Fujita's critical exponent,no matter how small the compactly supported initial data are.Moreover,we obtain an estimate of the lifespan when the power p satisfies:1<p<1 + 2/N.Secondly,we focus on the damped semilinear wave equations oU_tside the unit ball with Dirichlet boundary condition U_tt-?u+U_t=|u|p.Blow up result is established assuming that the power p of the nonlinear term |u|p satisfnes:1<p<1+2/N(N?3),which is known as Fujita's critical exponent,no matter how small the compactly supported initial data are.Moreover,we obtain an estimate of the lifespan when the power p satisfies:1<p<1 + 2/N.In the proving process,we mainly use the test function method.Finally,we are devoted to showing the polynomial decay estimates to the energy of dissipative wave equation with variable coefficients:(?),u =u(t,x)in(0,?)× RN with initial data u(0,x)=uo(x)and U_t(0,x)= u1(x).If the initial data are compactly supported from energy space,then there exists an exterior domain(?)such that for large t?>0,(?)with m>0.And moreover,if(?).