Existence Of Small Data Solutions Research Articles

Nonlinear fractional damped wave equation on compact Lie groups

In this paper, we deal with the initial value fractional damped wave equation on G, a compact Lie group, with power-type nonlinearity. The aim of this manuscript is twofold. First, using the Fourier analysis on compact Lie groups, we prove a local in-time existence result in the energy space for the fractional damped wave equation on G. Moreover, a finite time blow-up result is established under certain conditions on the initial data. In the next part of the paper, we consider fractional wave equation with lower order terms, i.e., damping and mass with the same power type nonlinearity on compact Lie groups, and prove the global in-time existence of small data solutions in the energy evolution space.

Weakly Coupled System of Semi-Linear Fractional θ-Evolution Equations with Special Cauchy Conditions

In this paper, we consider a weakly system of fractional θ-evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system.

A note to semilinear de Sitter models in 1d with balanced mass and dissipation

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.

Fractional Nonlinearity for the Wave Equation with Friction and Viscoelastic Damping

In this paper we consider a fractional nonlinearity for the wave equation with friction and viscoelastic damping. Using Fixed point theorem a global in time existence of small data solutions to the Cauchy problem is investigated in this research. Our main interest is to show the influence of the fractional nonlinearity parameter to the admissible range of exponent ς comparing with power nonlinearity and also the generating of loss of decay.

Weakly Coupled Systems of Semilinear Damped Waves with Different Scale-Invariant Time-Dependent Dissipation Terms

In this paper, our interest is the global existence of small data solutions to the Cauchy problem for weakly coupled systems of semilinear damped waves with different scale-invariant dissipation terms, where the data are supposed to belong to different classes of regularity and different power nonlinearities.

Semilinear effectively damped wave models with general relaxation function

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for semilinear effective damped wave models with general relaxation function in the source term. Our aim is to generalize some known results for special nonlinear memory terms, where the convolution is given with respect to the time variable. We first present auxiliary estimates for integrals. Then we prove results on global (in time) existence of small data Sobolev solutions for different classes of data. We distinguish between low regular data, data with suitable higher regularity and large regular data.

A Global Existence Result for a Semilinear Wave Equation with Lower Order Terms on Compact Lie Groups

In this paper, we study the semilinear wave equation with lower order terms (damping and mass) and with power type nonlinearity \(|u|^p\) on compact Lie groups. We will prove the global in time existence of small data solutions in the evolution energy space without requiring any lower bounds for \(p>1\). In our approach, we employ some results from Fourier analysis on compact Lie groups.

The critical exponent for semilinear [formula omitted]-evolution equations with a strong non-effective damping

In this paper, we find the critical exponent for the existence of global small data solutions to: utt+(−Δ)σu+(−Δ)θ2ut=f(u,ut),t≥0,x∈Rn,(u,ut)(0,x)=(0,u1(x)),in the case of so-called non-effective damping, θ∈(σ,2σ], where σ≠1 and f=|u|α or f=|ut|α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α>α̃ and do not exist, in general, for subcritical powers 1<α<α̃. Assuming initial data to be small in L1 or in some other Lp space, p∈(1,2), in addition to the energy space, the critical exponent only depends on the ratio n/(σp). We also prove the global existence of small data solutions in high space dimension for α>ᾱ, but we leave open to determine if a counterpart nonexistence result for α<ᾱ holds or not.

The interplay of critical regularity of nonlinearities in a weakly coupled system of semi-linear damped wave equations

We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the p − q plane. The main goal of this paper is to find out sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of (even) small data solutions.

A test function method for evolution equations with fractional powers of the Laplace operator

In this paper, we discuss a test function method to obtain nonexistence of global-in-time solutions for higher order evolution equations with fractional derivatives and a power nonlinearity, under a sign condition on the initial data. In order to deal with fractional powers of the Laplace operator, we introduce a suitable test function and a suitable class of weak solutions. The optimality of the nonexistence result provided is guaranteed by both scaling arguments and counterexamples. In particular, our manuscript provides the counterpart of nonexistence for several recent results of global-in-time existence of small data solutions to the following problem: utt+(−Δ)θut+(−Δ)σu=f(u,ut),t>0,x∈Rn,u(0,x)=u0(x),ut(0,x)=u1(x),with f=|u|p or f=|ut|p, where θ≥0 and σ>0 are fractional powers.

Global Existence of Small Data Solutions to Semi-linear Fractional $$\sigma $$-Evolution Equations with Mass and Nonlinear Memory

In this paper we study the global (in time) existence of small data solutions to semi-linear fractional $$\sigma -$$ evolution equations with mass and nonlinear memory. Our main goals was to explain on the one hand the influence of the memory term and on the other hand, the influence of higher regularity of the data on qualitative properties of solutions.

A structurally damped σ‐evolution equation with nonlinear memory

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: with σ&gt;0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m&gt;1.

Influence of strong time-dependent oscillations in semilinear damped wave models

We study the global (in time) existence of small data solutions to some Cauchy problems for semilinear damped wave models with strong time-dependent oscillations. The goal is to understand the influence of strong oscillations in the coefficients on solutions to some semilinear models with an “effective-like” damping term, where the data are supposed to belong to different classes of regularity.

Existence of global solutions for a weakly coupled system of semilinear viscoelastic damped $\sigma$-evolution equations

We prove the global (in time) existence of small data solutions from energy spaces based on L q spaces, q ∈ ( 1 , ∞ ) , to the Cauchy problem for a weakly coupled system of semilinear viscoelastic damped σ -evolution equations, where we consider nonlinearity terms with powers p 1 , p 2 > 1 and any σ 1 , σ 2 ≥ 1 in the comparison between two single equations. To do this, by mixing additional L m regularity for the data on the basis of L q - L q estimates, with q ∈ ( 1 , ∞ ) and m ∈ [ 1 , q ) , we apply ( L m ∩ L q ) - L q estimates for solutions to the corresponding linear Cauchy problems to treat semilinear problems. In addition, two different strategies allowing no loss of decay and some loss of decay combined with the flexible choice of admissible parameters σ 1 , σ 2 , m and q bring some benefits to relax the restrictions on the admissible exponents p 1 , p 2 .

Decay properties and asymptotic profiles for elastic waves with Kelvin–Voigt damping in 2D

In this paper we consider elastic waves with Kelvin-Voigt damping in 2D. For the linear problem, applying pointwise estimates of the partial Fourier transform of solutions in the Fourier space and asymptotic expansions of eigenvalues and their eigenprojections, we obtain sharp energy decay estimates with additional $L^m$ regularity and $L^p-L^q$ estimates on the conjugate line. Furthermore, we derive asymptotic profiles of solutions under different assumptions of initial data. For the semilinear problem, we use the derived $L^2-L^2$ estimates with additional $L^m$ regularity to prove global (in time) existence of small data solutions to the weakly coupled system. Finally, to deal with elastic waves with Kelvin-Voigt damping in 3D, we apply the Helmholtz decomposition.

Semilinear wave models with friction and viscoelastic damping

In this paper, we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear wave equation with friction, viscoelastic damping, and a power nonlinearity. We are interested in the connection between regularity assumptions for the data and the admissible range of exponents p in the power nonlinearity.

Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power non-linearity. We prove that the critical exponent is the Fujita exponent pFuj(Q)=1+2/Q, where Q is the homogeneous dimension of the Heisenberg group.On the one hand, we will prove the global existence of small data solutions for p>pFuj(Q) in an exponential weighted energy space. On the other hand, a blow-up result for 1<p≤pFuj(Q) under certain integral sign assumptions for the Cauchy data by using the test function method.

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

In this note, we consider a semilinear wave equation with scale-invariant mass and dissipation. The scale of the dissipation is the same of the mass term and this creates an interplay in which a relation between the coefficients comes into play to determine if the dissipation is dominant with respect to the mass, or viceversa. We prove global existence of small data solutions for supercritical power nonlinearities when the mass term is dominant with respect to the dissipative term (“Klein–Gordon type” case). The critical exponent is a modified Strauss exponent for global small data solutions to semilinear Klein–Gordon equations.

Critical regularity of nonlinearities in semilinear classical damped wave equations

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.

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: urn:x-wiley:mma:media:mma5370:mma5370-math-0001 where with b2 &gt; a2 &gt; 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p1,p2,p3 in to get results for the global (in time) existence of small data solutions.