On potential wells and applications to semilinear hyperbolic equations and parabolic equations Original Research Article

In this paper we generalize the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations and parabolic equations View the MathML sourceutt-Δu=f(u),x∈Ω,t>0, Turn MathJax on View the MathML sourceu(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω, Turn MathJax on View the MathML sourceu(x,t)=0,x∈∂Ω,t⩾0 Turn MathJax on and View the MathML sourceut-Δu=f(u),x∈Ω,t>0, Turn MathJax on View the MathML sourceu(x,0)=u0(x),x∈Ω, Turn MathJax on View the MathML sourceu(x,t)=0,x∈∂Ω,t⩾0, Turn MathJax on not only give a threshold result of global existence and nonexistence of solutions, but also obtain the vacuum isolating of solutions. Finally we prove the global existence of solutions for above problem with critical initial conditions I(u0)⩾0I(u0)⩾0, E(0)=dE(0)=d or I(u0)⩾0I(u0)⩾0, J(u0)=dJ(u0)=d. So Payne and Sattingerʹs results are generalized and improved in essential.