In this paper, we study the Cauchy problem for the linear plate equation with mass term and its applications to semilinear models. For the linear problem, we obtain L^p-L^q estimates for the solutions in the full range 1≤ p≤ q≤∞ , and we show that such estimates are optimal. In the sequel, we discuss the global in time existence of solutions to the associated semilinear problem with power nonlinearity |u|^α . Assuming small initial data in H^2(ℝ^n)× L^2(ℝ^n) , the presence of the mass term allows us to obtain global in time existence of energy solutions for all 1<α≤ (n+4)/[n-4]_+ . We show that the latter upper bound is optimal, since we prove that there exist data such that a non-existence result for local weak solutions holds when α > (n+4)/[n-4]_+ . Moreover, we study the influence on the long-time behaviour of solutions under the additional assumption of data in L^p(ℝ^n) , with p∈ [1, 2) . More precisely, we prove that for α >1 + 4p/n the solutions to the semilinear problem have the same long-time behaviour as the solutions to the linear problem.

In this paper, we study the Cauchy problem for the linear plate equation with mass term and its applications to semilinear models. For the linear problem, we obtain L-p-L-q estimates for the solutions in the full range 1 <= p <= q <= infinity, and we show that such estimates are optimal. In the sequel, we discuss the global in time existence of solutions to the associated semilinear problem with power nonlinearity |u|(alpha).Assuming small initial data in H-2([R-n) x L-2([R-n), the presence of the mass term allows us to obtain global in time existence of energy solutions for all 1 (n + 4)/[n-4](+).Moreover, we study the influence on the long-time behaviour of solutions under the additional assumption of data in L-p([R-n), with p is an element of [1, 2). More precisely, we prove that for alpha > 1 + (4n)/(p) the solutions to the semilinear problem have the same long-time behaviour as the solutions to the linear problem.

In this paper we derive sharp L (p) - L-q estimates, 1 <= p <= q <= infinity (including endpoint estimates as L-1 - L-1 and L-1 - L-infinity) for dissipative wave-type equations, under the assumption that the dissipation dampen the oscillations but it does not cancel them. We assume that the phase function w is homogeneous of some degree sigma > 0 and that its Hessian matrix has maximal rank, including the critical case sigma = 1, while the dissipative term a(xi ) > 0 may be inhomogeneous. The critical case includes waves with viscoelastic or structural damping, damped double dispersion equations and plate equations with rotational inertia, and so on. We also obtain the analogous results for fractional Schr & ouml;dinger-type equations with a potential.

In this paper, we derive suitable optimal L^p-L^q decay estimates, 1≤ p≤ 2≤ q≤∞ , for the solutions to the σ -evolution equation, σ >1 , with scale-invariant time-dependent damping and power nonlinearity |u|^p , u_tt+(-Δ )^σ u + μ/1+t u_t= |u|^p, t≥ 0, x∈ℝ^n, where μ >0 , p>1 . The critical exponent p=p_c for the global (in time) existence of small data solutions to the Cauchy problem is related to the long time behavior of solutions, which changes accordingly μ∈ (0, 1) or μ >1 . Under the assumption of small initial data in L^m(ℝ^n)∩ L^2(ℝ^n), m=1,2 , we find the critical exponent at low space dimension n with respect to σ , namely, p_c= max{p̅(γ _m), p̅ (γ _m+μ -1) } , γ _m := n/mσ, μ >1-γ _m, where p̅(γ ) := 1+ 2/γ is the well known Fujita exponent. Hence, p_c=p̅(γ _m) if μ >1 , whereas p_c=p̅ (γ _m+μ -1) is a shift of Fujita type exponent if μ∈ (0, 1) .

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

In this paper we consider the Cauchy problem for semilinear de Sitter models in 1d with balanced mass and dissipation. The model of interest is ϕtt−e−2tϕxx+ϕt+14ϕ=|ϕ|p,(ϕ(0,x),ϕt(0,x))=(0,g(x)).We study the global (in time) existence of small data solutions. In particular, we show by applying Schauder’s fixed point theorem that there exists a Sobolev solution for all p>1.

In this paper, we consider the Cauchy problem for scale-invariant semilinear wave models with time-dependent mass, dissipation and integrable time-dependent propagation speed. The goal is to study the interplay between the coefficients appearing in the mass and dissipation term and the exponent in the speed of propagation to prove global (in time) existence of small data Sobolev solutions and blow-up results.

In the present paper, we study the influence of oscillations of the time-dependent damping term b(t)u(t) on the asymptotic behavior of the energy for solutions to the Cauchy problem for a s-evolution equation {u(tt) + (- Delta)(sigma) u + b(t)u(t) = 0, (t, x) is an element of [0,for all) x R-n, u(0, x) = u(0)(x), ut (0, x) = u(1)(x), x is an element of R-n, where sigma > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b(t) = beta(1 + t)(-1), with beta > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of beta.

In this paper, we find the critical exponent for the existence of global small data solutions to: {u(tt) + (-Delta)(sigma)u + (-Delta)(theta/2) u(t) = f(u, u(t)), t >= 0, x epsilon R-n, (u, u(t))(0, x) = (0, u(1)(x)), in the case of so-called non-effective damping, theta epsilon (sigma, 2 sigma Sigma], where sigma not equal 1 and f = vertical bar u vertical bar(alpha) or f = vertical bar u(t)vertical bar(alpha), in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers alpha > (alpha) over tilde and do not exist, in general, for subcritical powers 1 < alpha < (alpha) over tilde. Assuming initial data to be small in L-1 or in some other Lp space, p epsilon (1, 2), in addition to the energy space, the critical exponent only depends on the ratio n/(Sigma p). We also prove the global existence of small data solutions in high space dimension for alpha > (alpha) over bar, but we leave open to determine if a counterpart nonexistence result for alpha < <(alpha)over tilde> holds or not. (C)2021 Published by Elsevier Ltd.

In this paper, we derive long time L-p - L-q decay estimates, in the full range 1 <= p <= q <= infinity, for time-dependent multipliers in which an interplay between an oscillatory component and a diffusive component with different scaling appears. We estimate parallel to m(t, .)parallel to M-p(q) as t -> infinity for multipliers of type m(t, xi) = e(+/- i vertical bar xi vertical bar sigma t-vertical bar xi vertical bar theta t), and suitable perturbations, under the assumption that the scaling of the diffusive component is worse, i.e., theta > sigma. These multipliers are, for instance, related to the fundamental solution to the Cauchy problem for the sigma-evolution equation with structural damping: u(tt) + (-Delta)(sigma)u + (-Delta)(theta/2) u(t) = 0, t >= 0, x is an element of R-n, in the so-called non-effective case sigma < theta. (C) 2021 Elsevier Inc. All rights reserved.

We consider the nonlinear massless wave equation belonging to some family of the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

In this paper we study the asymptotic profile (as t -> infinity) of the solution to the Cauchy problem for the linear plate equation u(tt) + Delta(2)u - lambda(t)Delta u + u(t) = 0 when lambda = lambda(t) is a decreasing function, assuming initial data in the energy space and verifying a moment condition. For sufficiently small data, we find the critical exponent for global solutions to the corresponding problem with power nonlinearity u(tt) + Delta(2)u - lambda(t)Delta u + u(t) = vertical bar u vertical bar(p). In order to do that, we assume small data in the energy space and, possibly, in L-1. In this latter case, we also determinate the asymptotic profile of the solution to the semilinear problem for supercritical power nonlinearities.

In this paper, we derive suitable optimal L^p-L^q decay estimates, 1≤ p≤ q≤∞, for the solutions to the σ-evolution equation, σ>1, with structural damping and power nonlinearity |u|^1+α or |u_t|^1+α, u_tt+(-Δ)^σ u +(-Δ)^θ u_t= |u|^1+α, |u_t|^1+α, where t≥0 and x∈ℝ^n. Using these estimates, we can solve the problem of finding the critical exponents for the two nonlinear problems above in the so-called non-effective case, θ∈(σ/2,σ]. This latter is more difficult than the effective case θ∈[0,σ/2), since the asymptotic profile of the solution involves a diffusive component and an oscillating one. The novel idea in this paper consists in treating separately the two components to neglect the loss of decay rate created by the interplay of the two components. We deal with the oscillating component, by localizing the low frequencies, where oscillations appear, in the extended phase space. This strategy allows us to recover a quasi-scaling property which replaces the lack of homogeneity of the equation.

We consider the Cauchy problem on $$\mathbf {R_{0}^+} \times \mathbf {R^n}$$ for the semilinear damped wave equation $$\displaystyle u_{tt}(t,x) - a^2(t) \Delta u(t,x) + b(t) u_t(t,x)= |u(t,x)|{ }^p $$ with decreasing in time coefficients, the propagation speed a(t) = (1 + t)−ℓ, ℓ ∈ (0, 1), the scale-invariant dissipation b(t) = β(1 + t)−1, β > 0, and a power nonlinearity of order p > 1. The solution u 0 of the corresponding linear Cauchy problem will be represented in the explicit form using Fourier multipliers operators with multipliers expressed in terms of special functions. Our main goal is to prove a global in time existence result when initial data belongs to the space H m(R n) × H m−1(R n), m ≥ 1. We are focused in finding the critical exponent p c(n, ℓ) such that if 1 < p < p c(n, ℓ) there exist small data for which u blow-up in finite time. We also prove that if p ≥ p c(n, ℓ) the global solution has the same long time behavior as u 0. In order to estimate u we use Duhamel’s principle to represent u and then we apply L 2 − L 2 estimates of u 0.

In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % \[ u_{tt}+(-\Delta)^\delta u_{tt}+(-\Delta)^\alpha u+(-\Delta)^\theta u_t=|u_t|^p, \quad t\geq 0,\,\, x\in\R^n,\] % with $p>1$, $2\theta \in [0, \alpha]$ and $\delta \in (\theta,\alpha]$. We show that, under additional regularity $\left(H^{\alpha+\delta}(\R^n)\cap L^{m}(\R^n) \right)\times \left(H^{2\delta}(\R^n)\cap L^{m}(\R^n)\right) $ for initial data, with $m\in (1,2]$, the critical exponent is given by $p_c=1+\frac{2m\theta}{n}$. The nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters $\alpha, \delta, \theta$, by using the test function method (under suitable sign assumptions on the initial data).

In this paper we consider the Cauchy problem for the semilinear damped wave equation $$\begin{aligned} u_{tt} - {\varDelta }u + u_t= h(u), \quad u(0,x)=\phi (x), \quad u_t(0,x)= \psi (x), \end{aligned}$$where $$h(s)=|s|^{1+ \frac{2}{n}}\mu (|s|)$$. Here n is the space dimension and $$\mu $$ is a modulus of continuity. Our goal is to obtain sharp conditions on $$\mu $$ to obtain a threshold between global (in time) existence of small data solutions (stability of the zero solution) and blow-up behavior even of small data solutions.

In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation with Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional. To prove our result, we first derive Lr-Lq linear estimates for the solution to the inhomogeneous linear Cauchy problem and then we apply a contraction argument.