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

Abstract

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

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

“…Among these approaches we find, on the one hand, comparison arguments based either on the fundamental solution for the operator ∂ 2 t − t 2ℓ ∆ in [16] or on the employment of a special positive solution of the corresponding homogeneous equation involving modified Bessel functions of second kind in [10], and, on the other hand, a modified test function method in [15]. In particular, the result that we are going to show for (5) is consistent with those for (6) in [15,10], although we shall use a different approach to prove this result. Moreover, we will see that the upper bound for the exponent p in the blow-up result for (1) is a shift of the corresponding upper bound for (5).…”

“…has a crucial role in determining some properties of the fundamental solution of L k,µ,ν 2 . In the special case k = 0 (the so-called wave operator with scale-invariant damping and mass), it is known in the literature that the value of δ affects not only the fundamental solution of L 0,µ,ν 2 but also the critical exponents in the treatment of semilinear Cauchy problem associated with L 0,µ,ν 2 with power nonlinearity [18,25,21,19,20,26,3,2], nonlinearity of derivative type [27,8], and combined nonlinearity [6,7,9]. We shall see that even in the case k ∈ (0, 1) some properties of the fundamental solution of L k,µ,ν 2 depend strongly on the value of δ.…”

Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type

“…Hence, the solution v(x, τ ) verifies the following equation: 2) . Moreover, thanks to the above transformation, which implies a kind of similarity with the scale-invariant damping case, we can inherit the methods used in some previous works [1,8,9,10,11] to build the proof of our main result.…”

Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities

“…Note that the equation (1.10) is somehow related to the Euler-Darboux-Poisson equation. Moreover, thanks to this transformation, which implies a kind of similarity with the scale-invariant damping case, we can inherit the methods used in our previous works [4,5,6] to build the proofs of our main results which are related, as a first target, to the blow-up of the solution of (1.1) and, as a secondary aim, to the blow-up of (1.7).…”