A Note on $$L^{p}-L^{q}$$ Estimates for Semilinear Critical Dissipative Klein–Gordon Equations

Abstract

In this paper we derive L p − L q estimates, with 1 ≤ p ≤ q ≤ ∞ (including endpoint estimates asfor a general class of dissipation terms, where Af = F −1 (a(ξ) F f (ξ)), with a ∈ C n+1 (R n \ {0}), and a(ξ) > 0 verifies conditions of Mikhlin-Hörmander type for M q p multipliers which may be different at low frequencies and at high frequencies; in particular a(ξ) may also be inhomogeneous and anisotropic. We prove that the obtained estimates are sharp.  Key words and phrases. Dissipative wave equations, L p − L q estimates, M q p multipliers, decay estimates, non-effective damping.

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

“…In any case, this is not an issue as in the global existence part one would deal in this specific case with a strict lower bound for the exponents. A similar situation is present in [7] in the case δ = 0 but for study of the semilinear Cauchy problem with power nonlinearity (3).…”

“…Let us recall the meaning of critical curve for a weakly coupled system: if the exponents p, q > 1 satisfy max{Λ(n, p, q), Λ(n, q, p)} < 0 (supercritical case), then, it is possible to prove a global existence result for small data solutions; on the contrary, for max{Λ(n, p, q), Λ(n, q, p)} ≥ 0 it is possible to prove the nonexistence of global in time solutions regardless the smallness of the Cauchy data and under certain sign assumptions for them. Let us point out that, according to the results we quoted above, the conjecture that the critical line for (6) is given by (7) has be shown to be true only partially, as the global existence of small data solutions has been proved only in the 3-dimensional and radial symmetric case. In the massless case (ν 2 1 = ν 2 2 = 0) and scattering producing case, that is, if we consider time-dependent, nonnegative and summable coefficients b 1 (t), b 2 (t) instead of µ 1 (1 + t) −1 , µ 2 (1 + t) −1 , really recently in [42] a blow-up result has been proved in the same range for the exponents (p, q) as for the corresponding not damped case, namely, for (p, q) such that max{Λ(n, p, q), Λ(n, q, p)} ≥ 0 is satisfied.…”

A global existence result for a semilinear scale‐invariant wave equation in even dimension

“…For further considerations on how the quantity describes the interplay between the damping term 1+t u t and the mass term 2 (1+t) 2 u one can see. 14 Recently, (1.3) has been studied in D'Abbicco and Palmieri, Nunes do Nascimento et al, Palmieri, and Palmieri and Reissig [15][16][17][18][19][20] under different assumptions on .…”

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

“…R n |u(s, x)| q ψ(s, x) dx ds (7) for any φ, ψ ∈ C ∞ 0 ([0, T ) × R n ) and any t ∈ [0, T ). After a further integration by parts in (6) and (7), letting t → T , we find that (u, v) fulfills the definition of weak solution to (1).…”

“…homogeneous linear equation. According to [3,40,5,4,39,22,30,27,21,13,31,37,38,28,29,6,35,16,20] for δ 0 the model in (2) is somehow an intermediate model between the semilinear free wave equation and the semilinear classical damped equation, whose critical exponent is p Fuj (n + µ−1 2 − √ δ 2 ) for δ ≥ (n + 1) 2 and seems reasonably to be p 0 (n + µ) for small and nonnegative values of delta, where p Fuj (n) and p 0 (n) denote the Fujita exponent and the Strauss exponent, respectively.…”