In this paper, we investigate the semilinear equation with a time-space fractional structural damping and a nonlocal in time nonlinearity \begin{equation*} {\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^σu+(-Δ)^δ\mathbf{D}_{0|t}^{α_2} u = I_{0|t}^{1-γ}|u|^{p}, \qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \end{equation*} where $p>1$, $α_i, γ in (0,1)$, $δ, σ\in (0,1)$, ${\mathbf{D}}_{0|t}^{α_i}$ is the Caputo fractional derivative and $I_{0|t}^{1-γ}$ is the Riemann-Liouville fractional integral of order $1-γ$. We prove the non-existence of global solutions if \begin{equation*} 1<p\leqslant \frac{2(2+α_1-γ)}{(\frac{α_1+1}σ N+2γ-2α_1-2)_+ }+1, \end{equation*} for any space dimension $N\geqslant 1.$ Then, we extend the result to the system \begin{align*} &{\mathbf{D}}_{0|t}^{1+α_1}u + (-Δ)^{σ_1} u + (-Δ)^{δ_1}{\mathbf{D}}_{0|t}^{α_2} u = I_{0|t}^{1-γ_{1}}|v|^{p},\qquad (t,x)\in (0,\infty) \times \mathbb{R}^N, \\ &{\mathbf{D}}_{0|t}^{1+β_{1}}v+(-Δ)^{σ_2} v + (-Δ)^{δ_2}{\mathbf{D}}_{0|t}^{β_2}v = I_{0|t}^{1-γ_2}|u|^{q},\qquad (t,x)\in (0,\infty )\times \mathbb{R}^{N}, \end{align*} where $p,q>1$, $0<δ_i$, $σ_i<1$ and $γ_2\in (0,1)$. Also, we present the necessary conditions for the existence of local or global solutions.
In this paper we consider the following biharmonic equation with critical exponent $P_ε$ : $Δ^2 u= Ku^{(n+4)/(n-4)-ε}, u>0$ in $Ω$ and $u=Δu=0$ on $\partialΩ$, where $Ω$ is a domain in $R^n$, $n\geq 5$, $ε$ is a small positive parameter and $K$ is smooth positive function. We construct solutions of $P_ε$ which blow up and concentrate at strict local maximum of $K$ either at the boundary or in the interior of $Ω$. We also construct solutions of $P_ε$ concentrating at an interior strict local minimum of $K$. Finally, we prove a nonexistense result for the corresponding supercritical problem whic…
In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence.
In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Using the the Faedo-Galerkin method we establish the global existence of the solution of the problem and we also prove few general decay rate results.
In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalised Young's inequality are mainly used to prove the decay estimate.
We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.
In this paper, we study the long-time dynamics for the wave equation with nonlocal weak damping and sup-cubic nonlinearity in a bounded smooth domain of $\mathbb{R}^3.$ Based on the Strichartz estimates for the case of bounded domains, we first prove the global well-posedness of the Shatah-Struwe solutions. Then we establish the existence of the global attractor for the Shatah-Struwe solution semigroup by the method of contractive function. Finally, we verify the existence of a polynomial attractor for this semigroup.
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
Let $Ω$ be a bounded smooth domain in $\mathbb{R}^{4}$ such that for some integer $d\geq1$ its $d$-th singular cohomology group with coefficients in some field is not zero, then problem {Δ^{2}u-ρ^{4}k(x)e^{u}=0 & \hbox{in}Ω, u=Δu=0 & \hbox{on}\partialΩ, has a solution blowing-up, as $ρ\to0$, at $m$ points of $Ω$, for any given number $m$.
An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed, thanks to a better understanding of the dynamics of the solutions, we give here a better characterization of the blow-up region. Furthermore, the techniques used in this article may be extended to other systems and interestingly they simplify the proof of the blow-up result in [3] which is concerned with the single wave equation in the same context as in th…
Blow-up and lifespan estimate for the generalized tricomi equation with the scale-invariant damping and time derivative nonlinearity on exterior domain
In this paper, we first establish a criterion based on contractive function for the existence of polynomial attractors. This criterion only involves some rather weak compactness associated with the repeated limit inferior and requires no compactness, which makes it suitable for critical cases. Then by this abstract theorem, we verify the existence of a polynomial attractor and estimate its attractive speed for the wave equation with nonlocal weak damping, anti-damping and critical nonlinearity.
We consider in this article the weakly coupled system of wave equations in the \textit{scale-invariant case} and with time-derivative nonlinearities. Under the usual assumption of small initial data, we obtain an improvement of the delimitation of the blow-up region by obtaining a new candidate for the critical curve. More precisely, we enhance the results obtained in \cite{Palmieri} for the system under consideration in the present work. We believe that our result is optimal in the sense that beyond the blow-up region obtained here we may conjecture the global existence of the solution.
In this article, we consider the damped wave equation in the \textit{scale-invariant case} with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: $$ (E) \quad u_{tt}-t^{2m}Δu+\fracμ{t}u_t+\frac{ν^2}{t^2}u=|u_t|^p, \quad \mbox{in}\ \mathbb{R}^N\times[1,\infty), $$ that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, $ν$ and $μ>0$, respectively, that the blow-up region and the lifespan bound of the solution of $(E)$ remain the same …
In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. I…
In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-…
In this paper we consider the critical exponent problem for the semilinear wave equation with space-time dependent damping. When the damping is effective, it is expected that the critical exponent agrees with that of only space dependent coefficient case. We shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent. Moreover, we do not assume that the data are compactly supported. However, it is still open whether there exists a blow-up solution if the power of nonlinearity is smaller than the expected exponent.
Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity and a scale-invariant damping
University of Tunis El Manar（突尼斯曼纳尔大学）；Imam Abdulrahman Bin Faisal University（伊玛目阿卜杜勒拉赫曼·本·费萨尔大学）；Musashino University（武藏野大学）；Université Sorbonne Paris Nord（北索邦巴黎大学）
We consider Cauchy problem for the semilinear plate equation with nonlocal nonlinearity. Under mild conditions on the damping coefficient, we prove that the semigroup generated by this problem possesses a global attractor.
We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. …
In this paper, we consider an inverse problem for a nonlinear wave equation with a damping term and a general nonlinear term. This problem arises in nonlinear acoustic imaging and has applications in medical imaging and other fields. The propagation of ultrasound waves can be modeled by a quasilinear wave equation with a damping term. We show the boundary measurements encoded in the Dirichlet-to-Neumann map (DN map) determine the damping term and the nonlinearity at the same time. In a more general setting, we consider a quasilinear wave equation with a one-form (a first-order term) and a gene…
We consider the initial boundary value problem in exterior domain for strongly damped wave equations with power type nonlinearity |u|^p. We will establish blow-up results under some conditions on the initial data and the exponent p.
We consider the linear wave equation with the time-dependent scale-invariant damping and mass. We also treat the corresponding equation with the energy critical nonlinearity. Our aim is to show that the solution scatters to a modified linear wave solution and to obtain its asymptotic order.
In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. Our aim is to obtain the threshold, to classify the global existence of solution for small data or the finite time blow-up pf the solution, with respect to the growth order of the nonlinearity.
The initial value problem is considered for a higher order nonlinear Schrödinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional damping large-time decay of solutions without any smallness assumptions is also established. The main difficulty of the study is the non-smooth character of the nonlinearity.
In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients.
We present a notion of weak solution for the Dirichlet problem driven by the fractional Laplacian, following the Stampacchia theory. Then, we study semilinear problems on bounded domains $Ω$ with two different boundary conditions at the same time: the shape of the solution outside $Ω$ and a weighted limit to the boundary. We allow the nonlinearity to be positive or negative and we look for solutions blowing up at the boundary. Our starting observation is the existence of s-harmonic functions which explode at the boundary: these will be used both as supersolutions in the case of negative nonlin…
We consider the initial boundary value problem in exterior domain for semilinear wave equations with power-type nonlinearity |u| p. We will establish blow-up results when p is less than or equal to Strauss' exponent which is the same one for the whole space case R n .
Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.
We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.
This article is concerned with the asymptotic behavior of the two-dimensional inviscid Boussinesq equations with a damping term in the velocity equation. Precisely, we provide the time-decay rates of the smooth solutions to that system. The key ingredient is a careful analysis of the Green kernel of the linearized problem in Fourier space, combined with bilinear estimates and interpolation inequalities for handling the nonlinearity.
Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent
We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the whole energy-subcritical case. This result implies that small data blow-up does not occur in the overdamping case, different from the other cases, i.e. effective or non-effective damping.
The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^β}Δu_t$) and a nonlinearity $|u|^p$. We will show the influence of the the parameter $β$ in the blow-up results under some hypothesis on the initial data and the exponent $p$ by using the test function method. We also study the local existence in time of mild solution in the energy space $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$.
This paper is devoted to the lifespan of solutions to a damped fourth-order wave equation with logarithmic nonlinearity $$u_{tt}+Δ^2u-Δu-ωΔu_t+α(t)u_t=|u|^{p-2}u\ln|u|.$$ Finite time blow-up criteria for solutions at both lower and high initial energy levels are established, and an upper bound for the blow-up time is given for each case. Moreover, by constructing a new auxiliary functional and making full use of the strong damping term, a lower bound for the blow-up time is also derived.
In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in $\mathbb{R}^n$, which can be represented by the Riemann-Liouville fractional integral of order $1-γ$ with $γ\in(0,1)$. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of $γ$, on the blow-up range between the effective case and the non-effective ca…
In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.
In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is globally asymp-totically stable with a linear damping. To do so, we first characterize, in terms of Lyapunov functionals, several types of asymptotic stability for linear infinite-dimensional systems, namely the exponential and the polynomial stability. Then, we derive a Lyapunov functional for the nonlinear system, which is the sum of a Lyapunov functional c…
The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the \textit{scale-invariant case} and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($\fracμ{\sqrt{1+|x|^2}}u_t$), we provide that in higher dimensions the blow-up region is given by $p \in (1, p_G(N+μ)]$ where $p_G(N)$ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $μ$ given …
In this paper we consider the critical exponent problem for the semilinear damped wave equation with time-dependent coefficients. We treat the scale invariant cases. In this case the asymptotic behavior of the solution is very delicate and the size of coefficient plays an essential role. We shall prove that if the power of the nonlinearity is greater than the Fujita exponent, then there exists a unique global solution with small data, provided that the size of the coefficient is sufficiently large. We shall also prove some blow-up results even in the case that the coefficient is sufficiently s…
