Nonexistence of Global Solutions to A Semilinear Wave Equation with Scale Invariant Damping

We obtain a blowup result for solutions to a semilinear wave equation with scale-invariant dissipation. We perform a change of variables that transforms our starting equation into a Generalized Tricomi equation, then apply Kato’s lemma, we can prove a blowup result for solutions to the transformed equation under some assumptions on the initial data. In the critical case, we use the fundamental solutions of the Generalized Tricomi equation to modify Kato’s lemma to deal with it.

On the blowup of solutions of a Schrödinger equation with an inhomogeneous damping coefficient

We consider the Cauchy problem for the nonlinear self-focusing Schrödinger equation in ℝNwith an inhomogeneous smooth damping coefficient and we prove, for suitable initial data, and in the spirit of the seminal work of R. Glassey, a blowup result for the corresponding local solutions. We also give some lower bound estimates for the blowing-up solutions.

Blowup Phenomena for the Compressible Euler and Euler-Poisson Equations with Initial Functional Conditions

We study, in the radial symmetric case, the finite time life span of the compressible Euler or Euler-Poisson equations inRN. For timet≥0, we can define a functionalH(t)associated with the solution of the equations and some testing functionf. When the pressure functionPof the governing equations is of the formP=Kργ, whereρis the density function,Kis a constant, andγ>1, we can show that the nontrivialC1solutions with nonslip boundary condition will blow up in finite time ifH(0)satisfies some initial functional conditions defined by the integrals off. Examples of the testing functions includerN-1ln(r+1),rN-1er,rN-1(r3-3r2+3r+ε),rN-1sin((π/2)(r/R)), andrN-1sinh r. The corresponding blowup result for the 1-dimensional nonradial symmetric case is also given.

Blowup of solutions of a nonlinear wave equation

We establish a blowup result to an initial boundary value problem for the nonlinear wave equationutt−M(‖B1/2u‖ 2) Bu+kut=|u| p−2,x∈Ω,t>0.