Critical Exponent for a Nonlinear Wave Equation with Damping
نویسندگان
چکیده
It is well known that if the damping is missing, the critical exponent for the nonlinear wave equation gu=|u| p is the positive root p0(n) of the equation (n&1) p&(n+1) p&2=0, where n 2 is the space dimension (for p0(1)= , see Sideris [14]). The proof of this fact, known as Strauss' conjecture [17], took more than 20 years of effort, beginning with Glassey doi:10.1006 jdeq.2000.3933, available online at http: www.idealibrary.com on
منابع مشابه
Nonlinear Dissipative Wave Equations with Potential
We study the long time behavior of solutions of the wave equation with absorbtion |u|p−1u and variable damping V (x)ut, where V (x) ∼ V0|x|−α for large |x|, V0 > 0. For α ∈ [0, 1) and 1 < p < (n + 2)/(n − 2) we establish decay estimates of the energy, L2 and Lp+1 norm of solutions. We find three different regimes of decay of solutions depending on the exponent of the absorbtion term. We show th...
متن کاملRegularity and Scattering for the Wave Equation with a Critical Nonlinear Damping
We show that the nonlinear wave equation u + ut = 0 is globally well-posed in radially symmetric Sobolev spaces Hk rad(R 3) × Hk−1 rad (R 3) for all integers k > 2. This partially extends the well-posedness in Hk(R3) × Hk−1(R3) for all k ∈ [1, 2], established by Lions and Strauss [12]. As a consequence we obtain the global existence of C∞ solutions with radial C∞ 0 data. The regularity problem ...
متن کاملNonlinear-damping continuation of the nonlinear Schrödinger equation — A numerical study
We study the nonlinear-damping continuation of singular solutions of the critical and supercritical NLS. Our simulations suggest that for generic initial conditions that lead to collapse in the undamped NLS, the solution of the weakly-damped NLS iψt(t, x) + ∆ψ + |ψ |p−1ψ + iδ|ψ |q−1ψ = 0, 0 < δ ≪ 1, is highly asymmetric with respect to the singularity time, and the post-collapse defocusing velo...
متن کاملA ug 2 01 4 Kirchhoff equations with strong damping
We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the “elastic” operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space assuming only degenerate...
متن کاملAsymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R
The existence of a random attractor in H1(R3)×L2(R3) is proved for the damped semilinear stochastic wave equation defined on the entire space R 3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the rand...
متن کامل