Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

In this article, we study the blow-up of the damped wave equation in the scale-invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: u tt − Δu + 1 + t u t = |u t | p + |u| q , in R N × [0, ∞), with small initial data. For < N(q−1) 2 and ∈ (0, *), where * > 0 is depending on the nonlinearties' powers and the space dimension (* satisfies (q − 1) ((N + 2 * − 1)p − 2) = 4), we prove that the wave equation, in this case, behaves like the one without dissipation (= 0). Our result completes the previous studies in the case where the dissipation is given by (1+t) u t ; > 1, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term 1+t u t is important. KEYWORDS blow-up, nonlinear wave equations, scale-invariant damping MSC CLASSIFICATION 35L71; 35B44

show abstract

“…We refer the reader to other studies [15][16][17][18] for more details. Now, we focus on the case > 0.…”

Section: Introductionmentioning

confidence: 99%

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

Hamouda

Hamza

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Then, there exist positive constants C 0 and ε 0 such that if 0 < ε ≤ ε 0 , then the Cauchy problem (A.1) admits a unique global solution (u, v) satisfying [17,Proposition 9.1], we know that for n = 3, the condition p > 5/2 is sharp in general for global existence of small solutions to (A.1). Nonexistence of global, small solutions has been studied also in [12], [5] for systems similar to (A.1).…”

Section: Appendix Amentioning

confidence: 99%

“…By [17,Proposition 9.1], we know that for n = 3, the condition p > 5/2 is sharp in general for global existence of small solutions to (A.1). Nonexistence of global, small solutions has been studied also in [12], [5] for systems similar to (A.1).…”

Section: Model System Satisfying the Weak Null Conditionmentioning

confidence: 99%

Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

Hidano

Yokoyama

2018

International Mathematics Research Notices

View full text Add to dashboard Cite

We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah.The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as t → ∞, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type L 2 weighted space-time estimate for derivatives.

show abstract

“…We refer the reader to [5,9,11,32] for more details. Note that it is proven in [11] that, for p > p G and q > q S , the equality in (1.6) yields the global existence of the solution of (1.1) (with µ = 0 and (a, b) = (1, 1)) without going through the intermediate step of "almost global solution".…”

Section: Introductionmentioning

confidence: 99%

Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities

Hamouda¹,

Hamza²

2020

Preprint

View full text Add to dashboard Cite

We consider in this article the damped wave equation, in the scale-invariant case with combined two nonlinearities, which reads as follows:(E)with small initial data. Compared to our previous work [8], we show in this article that the first hypothesis on the damping coefficient µ, namely µ < N (q−1) 2, can be removed, and the second one can be extended from (0, µ * /2) to (0, µ * ) where µ * > 0 is solution of (q − 1) ((N + µ * − 1)p − 2) = 4. Indeed, owing to a better understanding of the influence of the damping term in the global dynamics of the solution, we think that this new interval for µ characterizes better the threshold between the blow-up and the global existence regions. Moreover, taking advantage of the techniques employed in the problem (E), we also improve the result in [17,23] in relationship with the Glassey conjecture for the solution of (E) without the nonlinear term |u| q . More precisely, we extend the blow-up region from p ∈ (1, p G (N + σ)], where σ is given by (1.7) below, to p ∈ (1, p G (N + µ)] giving thus a better estimate of the lifespan in this case.

show abstract

Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

Cited by 20 publications

References 19 publications

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities

Contact Info

Product

Resources

About