Critical thresholds for semilinear damped wave equations with Riesz potential power nonlinearity and initial data in pseudo-measure spaces

where γ ∈ [0, n), p > 1 and the initial data with ϵ > 0 is size parameter, while I γ denotes the Riesz potential represents the Riesz potential of order γ. For any f ∈ L 1 loc (R n ), the Riesz potential is defined by

where the normalization constant is given by A γ = Γ( n-γ 2 )

. More generally, the Riesz potential I γ can be interpreted as the inverse operator of the fractional Laplacian in the sense that (-∆) -γ 2 f (x) = I γ (f (x)). For a more comprehensive account of these fundamental properties of the Riesz potential, we refer the reader to [31,49].

The nonlinear term I γ (|u| p ) appearing in the problem (1.1) is inspired by the recent paper [15]. Our main purpose is to determine a critical exponent for the nonlinear Cauchy problem (1.1) with (u 0 , u 1 ) ∈ Y q × Y q equipping q ∈ 0, n 2 , in which pseudo-measure spaces Y q are defined by (see, for instance, [3])

where S ′ (R n ) is the space of tempered distributions and Y q (R n )-norm is given by ∥f ∥ Y q (R n ) := sup

The critical exponent means the threshold condition on the exponent p for global (in time) Sobolev solutions and blow-up of local (in time) weak solutions with small data. To be specific, under additional Y q assumptions for the initial data, the new critical exponent, which will be propositionosed for (1.1), is p crit (n, q, γ) := 1 + 2 + γ n -q with q ∈ 0, n 2 and γ ∈ [0, n) (1.3) for 1 ≤ n ≤ 4. Another objective of the present paper is to establish sharp upper and lower bound estimates for the lifespan of weak solutions to the semilinear Cauchy problem (1.1). Here, the lifespan T ϵ of a solution is defined as T ϵ := sup{T > 0: there exists a unique local (in time) solution u to problem (1.1) on [0, T ) corresponding to a fixed parameter ϵ > 0}. Assuming that the initial data belong to the homogeneous negative Sobolev space Y q , where q ∈ 0, n 2 and 1 + n -2q + 2γ

we establish the sharpness of the new lifespan estimates for the solution T ϵ , namely

To the best of the authors’ knowledge, the sharpness of lifespan estimates in the literature has only been achieved in the case where the initial data belong to the space L 1 (R n ), with the exponent p < p crit = 1 + 2 n . Therefore, our results in the space Y q (R n ) provide a genuinely new setting in which the sharpness of the lifespan can also be obtained. Now, we consider the corresponding linear problem of (1.1), that is,

For the linear damped wave equation (1.4), we also establish time decay estimates for the solution u(t, x) in the L 2 (R n ) norm as well as in the homogeneous Sobolev norm Ḣs (R n ), where s may be either negative or positive, assuming that the initial data satisfy u 0 , u 1 ∈ Y q . More precisely, these decay estimates are stated as follows:

2 -j ∥φ∥ H s+j-1 ∩Y q for j = 0, 1 and s > q -n 2 ,

where K(t, x) is defined as in Section 2.1. These spaces were originally introduced in harmonic analysis and later proved to be particularly useful in the study of dissipative partial differential equations. The quantity |ξ| q | f (ξ)| measures the behavior of f near the low-frequency region ξ = 0. Hence, the parameter q quantifies the strength of the singularity (or decay) of the Fourier transform at low frequencies. When q increases, stronger control near ξ = 0 is imposed. For this reason, Y q spaces are well adapted to problems where the long-time dynamics are governed by low-frequency behavior.

1.2. Background of this paper. In recent years, pseudo-measure spaces have played an important role in the study of incompressible fluid models. Let us recall the fractional Navier-Stokes equations in R 3 (see [38,2,11,7,1,48,17,37,51,28] and papers included):

where u(t, x) ∈ R 3 denotes the velocity field, π(x, t) ∈ R is the pressure, and α ∈ 0, 5 4 in the supercritical regime. The operator (-∆) α is defined via the Fourier transform by (see [47,49])

A large body of literature has been devoted to understanding temporal decay rates of weak solutions in L 2 R 3 and in negative Sobolev spaces Ḣ-s R 3 . More recently, these decay properties were refined by considering initial data in pseudo-measure spaces. In particular, in [36], it was shown that if u 0 ∈ L 2 R 3 ∩Y q R 3 , then the corresponding weak solution to the fractional Navier-Stokes equations (1.5) satisfies improved decay estimates of the form

4α , t > 0, together with analogous estimates in negative Sobolev norms. These results clearly demonstrate that the decay rate depends explicitly on the low-frequency index q. In other words, the long-time behavior of weak solutions is strongly influenced by the size of the Fourier transform near ξ = 0, which is precisely measured by the Y q norm. Such developments indicate that pseudo-measure spaces provide a natural and flexible framework for studying dissi