Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation below can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous blow up conjecture about critical semilinear wave equations of the form $\Delta u - \partial^2_t u + |u|^p = 0$ in dimensions $n \ge 4$. The lower dimensional case $n \le 3$ was settled many years earlier.

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n ≥ 4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

A shift in the Strauss exponent for semilinear wave equations with a not effective damping

Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, II

First we give a truly short proof of the major blowup result [T. C. Sideris, J. Differential Equations, 52 (1984), pp. 378--406] on higher-dimensional semilinear wave equations. Using this new method, we also establish blowup phenomenon for wave equations with a potential. This complements the recent interesting existence result by [V. Georgiev, C. Heiming, and H. Kubo, Comm. Partial Differential Equations, 26 (2001), pp. 2267--2303], where the blowup problem was left open.

https://www.math.ucr.edu/~qizhang/2004yz.pdf

Finite-Time Blowup for Wave Equations with a Potential

Blow-up of solutions to critical semilinear wave equations with variable coefficients

We study the asymptotic behavior of energy for wave equations with nonlinear damping g(u t ) = |u t | m-1 ut in R n (n ≥ 3) as time t → ∞ The main result shows a polynomial decay rate of energy under the condition 1 < m < (n + 2) /(n + 1) Previously, only logarithmic decay rates were found

Borislav Yordanov

Papers

Finite time blow up for critical wave equations in high dimensions

Finite time blow up for critical wave equations in high dimensions

Finite-Time Blowup for Wave Equations with a Potential

Blow-up of solutions to critical semilinear wave equations with variable coefficients

The energy decay problem for wave equations with nonlinear dissipative terms in Rn