Decay Estimates Of Global Solutions Research Articles

Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

Abstract This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) r\left(x) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω \omega -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.

Decay estimates for the time-fractional evolution equations with time-dependent coefficients

In this paper, the initial-boundary value problems for the time-fractional degenerate evolution equations are considered. Firstly, in the linear case, we obtain the optimal rates of decay estimates of the solutions. The decay estimates are also established for the time-fractional evolution equations with nonlinear operators such as: p-Laplacian, the porous medium operator, degenerate operator, mean curvature operator and Kirchhoff operator. At the end, some applications of the obtained results are given to derive the decay estimates of global solutions for the time-fractional Fisher-KPP-type equation and the time-fractional porous medium equation with the nonlinear source.

Note on a higher order pseudo‐parabolic equation with variable exponents

In this paper, we study a higher order pseudo‐parabolic equation involving ‐Laplacian with the Navier boundary condition. We use the energy method, the Sobolev embedding inequalities and the Galerkin's approximation to show the classification of singular solutions, including the existence and nonexistence of global, blow‐up, and extinction solutions. Extinction rates and time of extinction solutions, blow‐up time of blow‐up solutions, and decay estimates of global solutions are discussed.

Global existence and finite time blowup for a mixed pseudo-parabolic [formula omitted]-Laplacian type equation

This paper concerns the initial–boundary value problem for a mixed pseudo-parabolic p-Laplacian type equation. By constructing a family of potential wells, we first present the explicit expression for the depth of potential well, and then prove the existence, uniqueness and decay estimate of global solutions and the blowup phenomena of solutions with subcritical initial energy. Next, we extend parallelly these results to the critical initial energy. Lastly, the existence, uniqueness and asymptotic behavior of global solutions with supercritical initial energy are proved by further analyzing the properties of ω-limits of solutions.

Global existence and blow-up of solution to a class of fourth-order equation with singular potential and logarithmic nonlinearity

In this paper, we consider the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a fourth-order equation with strong damping and logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the method of Faedo–Galerkin approximation. By means of potential well method and Rellich inequality, we obtain the global existence and the decay estimate of global solutions under some appropriate conditions. Furthermore, we prove the finite time blow-up results of weak solutions, and establish the upper and lower bounds for blow-up time.

Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearity

In this paper, we consider a class of Boussinesq-type equations with logarithmic nonlinearity. By employing the classical Faedo–Galerkin method, we first establish the local well-posedness of solutions. Then we investigate the dynamical behaviors of solutions. More precisely, for the solutions with subcritical or critical initial energy, we prove that they exist globally and are uniformly bounded when I(u0)>0, where I(u0) denotes the Nehari functional with the initial value u0. Moreover, under further appropriate assumptions about the initial data, we derive the exponential energy decay estimates of global solutions. In particular, for the solutions with subcritical or critical initial energy, we show that they can be extended over time (the whole half line) and then blow up at infinite time when I(u0)<0. Last but not least, by developing some new methods, we prove the existence of infinite time blow-up solutions with arbitrary high initial energy.

Global Existence and Blow-up of Solutions for a System of Fractional Wave Equations

We investigate the Cauchy problem for a $2 \times 2$-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in $\mathbb{R}^{+} \times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the fractional orders of the time derivatives, it is shown that there exists a threshold value of the dimension $N$, for which, small data-global solutions as well as finite time blowing-up solutions exist. Furthermore, we investigate the $L^{\infty}$-decay estimates of global solutions.

Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p q ≤ ∞, and analyticity of solutions with 1 p q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

Dynamical boundary problem for Dirichlet-to-Neumann operator with critical Sobolev exponent and Hardy potential

We study the Laplacian equation with dynamical boundary condition involving Dirichlet-to-Neumann operator, critical growth, and Hardy potential. We first prove the existence and decay estimates of global solutions and finite time blowup of local solutions under certain assumptions. Then we focus on the asymptotic behavior of global solutions approaching a stationary solution in the long time series. Furthermore, we give a precise bubbling description by the concentration-compactness principle.

Global Existence and Blow-Up for the Fractional p-Laplacian with Logarithmic Nonlinearity

In this paper, we study the following Dirichlet problem for a parabolic equation involving fractional p-Laplacian with logarithmic nonlinearity $$\begin{aligned} \left\{ \begin{array}{llc} u_{t}+(-\Delta )^{s}_{p}u+|u|^{p-2}u=|u|^{p-2}u\log (|u|) &{} \text {in}\ &{} \Omega ,\;t>0 , \\ u =0 &{} \text {in} &{} {\mathbb {R}}^{N}\backslash \Omega ,\;t > 0, \\ u(x,0)=u_{0}(x), &{} \text {in} &{}\Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^N \, ( N\ge 1)$$ is a bounded domain with Lipschitz boundary and $$2\le p< \infty $$ . The local existence will be done using the Galerkin approximations. By combining the potential well theory with the Nehari manifold, we establish the existence of global solutions. Then by virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give decay estimates of global solutions. The main difficulty here is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.

Asymptotic profiles and convergence rate of the compressible fluid models of Korteweg type

In this paper, we consider the initial value problem for the compressible fluid models of Korteweg type in $${\mathbb {R}}^n(n\ge 3)$$ and asymptotic profile of global solutions and the corresponding convergence rate are established. The structure of the nonlinear term plays a very important role in constructing asymptotic profile. The proof is based on the decay estimate of solutions operator, decay estimate and weighted decay estimate of global solutions.

Global existence and exponential decay for a viscoelastic Petrovsky system

We investigate the initial-boundary value problem for a viscoelastic Petrovsky system of Kirchhoff-type with nonlinear source term. By applying the standard continuation principle, the existence of global solutions for this problem is proved and the exponential decay estimate of global solutions is obtained.

Decay estimate of global solutions to the generalized double dispersion model in Morrey spaces

In this paper, we investigate the initial value problem for the generalized double dispersion model in Morrey spaces. Based on the decay properties of the solution operator in Morrey spaces, global existence and decay estimates of solutions are proved by Banach fixed point theorem.

Global solutions and self‐similar solutions for coupled nonlinear Schrödinger equations

In this paper, we prove the existence and uniqueness for the global solutions of Cauchy problem for coupled nonlinear Schrödinger equations and obtain the continuous dependence result on the initial data and the stronger decay estimate of global solutions. In particular, we show the existence and uniqueness of self‐similar solutions. Also, we build some asymptotically self‐similar solutions. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Global existence and blow-up of solutions for a system of Petrovsky equations

The initial-boundary value problem for a system of Petrovsky equations with memory and nonlinear source terms in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set, and obtain the exponential decay estimate of global solutions. Meanwhile, under suitable conditions on relaxation functions and the positive initial energy as well as non-positive initial energy, it is proved that the solutions blow up in the finite time and the lifespan estimates of solutions are also given.

Large‐time behavior of solutions to the Rosenau equation with damped term

In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity‐loss type, which causes the difficulty in high‐frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for n≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for n≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for n = 1. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Global existence and asymptotic behavior of solutions for a system of higher-order Kirchhoff-type equations

This paper deals with the global existence and energy decay of solutions for some coupled system of higher-order Kirchhoff-type equations with nonlinear dissipa- tive and source terms in a bounded domain. We prove the existence of global solutions for this problem by constructing a stable set in H m1 0 (W) H m2 0 (W) and give the decay estimate of global solutions by applying a lemma of V. Komornik. with initial data u(x, 0) = u0(x), ut(x, 0) = u1(x), x2 W, (1.3) v(x, 0) = v0(x), vt(x, 0) = v1(x), x2 W,

Energy Decay of Global Solutions for a System of Petrovsky Equations

The initial-boundary value problem for a class of nonlinear Petrovsky systems in bounded domain is studied. We prove the energy decay estimate of global solutions through the use of a difference inequality.

Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions through the use of a difference inequality.

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

This paper gives the sufficient and necessary conditions of existence of global solutions and decay estimates of the solutions for the initial boundary value problem of some nonlinear parabolic equations with small initial energy and the nonlinear power less than Sobolev critical value. The existence, nonexistence and the decay estimates of global solutions are considered. The conditions that initial energy is small and nonlinear power is less than Sobolev critical value is imposed.