Initial Energy Case Research Articles

Well-posedness of solutions for a class of quasilinear wave equations with structural damping or strong damping

This paper deals with a class of quasilinear wave equations with structural damping or strong damping. By virtue of the improved Faedo–Galerkin method and some technical efforts, we first establish the local well-posedness of weak solutions, especially the continuity of weak solutions with respect to time in the natural phase space. Then we investigate the global existence, asymptotic behavior and finite time blow-up of weak solutions with subcritical or critical initial energy. As for the supercritical initial energy case, we show that the weak solutions may blow up in finite time with arbitrarily high initial energy. Last but not least, the upper and lower bounds of the blow-up time for blow-up solutions are derived.

Semilinear pseudo-parabolic equations on manifolds with conical singularities

<p style='text-indent:20px;'>This paper studies the well-posedness of the semilinear pseudo-parabolic equations on manifolds with conical degeneration. By employing the Galerkin method and performing energy estimates, we first establish the local-in-time well-posedness of the solution. Moreover, to reveal the relationship between the initial datum and the global-in-time well-posedness of the solution we divide the initial datum into three classes by the potential well depth, i.e., the sub-critical initial energy level, the critical initial energy level and the sup-critical initial energy level (included in the arbitrary high initial energy case), and finally we give an affirmative answer to the question whether the solution exists globally or not. For the sub-critical and critical initial energy, thanks to the potential well theory, we not only obtain the invariant manifolds, global existence and asymptotic behavior of solutions, but also prove the finite time blow up of solutions and estimate the lower bound the of blowup time. For the sup-critical case, we show the assumptions for initial datum which cause the finite time blowup of the solution, realized by introducing a new auxiliary function. Additionally, we also provide some results concerning the estimates of the upper bound of the blowup time in the sup-critical initial energy.</p>

Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations

Abstract In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.

Global existence and blow-up of solutions to a class of nonlocal parabolic equations

This paper is devoted to study a class of nonlocal parabolic equations, which was considered in Liu and Ma (2014) and Li and Liu (2017), where the case of initial energy J(u0)≤d (d is the mountain pass level) was discussed, the conditions on global existence or blow-up, the vacuum region and the asymptotic behavior of the solutions were studied. We extend their results on the following three aspects: Firstly, we explicitly give the vacuum region, the global existence region and the blow-up region when J(u0)<d, that is, there exist three regions U˜e, Ge and Be such that H01(Ω)=U˜e∪Ge∪Be, and1. U˜e is a vacuum region, i.e., the solution does not belong to U˜e;2. Ge is an invariant region, the solution exists globally and decays to 0 exponentially if the initial value belongs to Ge;3. Be is an invariant region, the solution blows up in finite time if the initial value belongs to Be.Secondly, we estimate the upper and lower bounds of the blow-up time and blow-up rate for the blow-up solutions when J(u0)≤d. Thirdly, we consider the asymptotic behavior of the solutions when J(u0)>d. By constructing two sets Ψα and Φα, we prove that the solution blows up in finite time if the initial value belongs to Ψα, while the solution exists globally and tends to zero as time t→+∞ when the initial value belongs to Φα.

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems (coupled parabolic systems) with nonlinear coupled source terms is considered in order to classify the initial data for the global existence, finite time blowup and long time decay of the solution. The whole study is conducted by considering three cases according to initial energy: the low initial energy case, critical initial energy case and high initial energy case. For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. In addition, for the high initial energy case the possibility of both global existence and finite time blowup is proved first, and then some sufficient initial conditions of finite time blowup and global existence are obtained, respectively.

Asymptotic behavior for a fourth-order parabolic equation involving the Hessian

This paper deals with a fourth-order parabolic PDE arising in the theory of epitaxial growth of crystal. For the stationary problem, we find a ground-state solution on its corresponding Nehari manifold by Lagrange multiplier method. As far as the evolution problem is concerned, we study the dynamics for both the global solution and the blow-up solution. Especially, for the global solution, we prove that the energy functional decays exponentially and we get the concretely decay rate. For the blow-up solution, the exponential growth of the solution is obtained, and we also obtain the growth rate. The behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. Moreover, we prove that there exists blow-up solution with arbitrary high initial energy. Finally, for the low initial energy case, we give some equivalent conditions for the solutions blow up in finite time and exist globally, respectively. Our results extend the results got by Escudero et al. (Eur J Appl Math 24(3):437–453, 2013; J Differ Equ 254(6):2515–2531, 2013; Math Method Appl Sci 37(6):793–807, 2014; J Math Pures Appl 103(4):924–957, 2015).

Global existence and finite time blow-up of the solution for a thin-film equation with high initial energy

This paper is devoted to the study of a thin-film equation, which was considered extensively in recent years. We obtain the conditions for global existence and finite time blow-up of solution for the high initial energy case, i.e., J(u0)>d, while the previous works only considered the case J(u0)≤d, here d is the mountain pass level. Moreover, for the blow-up solution with J(u0)<d, we obtain the lower bound estimate of the blow-up time and blow-up rate.

Global non-existence for some nonlinear wave equations with damping and source terms in an inhomogeneous medium

In this paper we study the initial value problem for the nonlinear wave equation with damping and source terms \t\t\tutt−ρ(x)−1Δu+ut+m2u=f(u)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}-\\rho(x)^{-1}\\varDelta u+u_{t}+m^{2}u=f(u) $$\\end{document} with some rho(x) and f(u) on the whole space mathbb{R}^{n} (ngeq 3).For the low initial energy case, which is the non-positive initial energy, based on a concavity argument we prove the blow-up result. As for the high initial energy case, we give sufficient conditions of the initial data such that the corresponding solution blows up in finite time. In other words, our results imply a complete blow-up theorem in the sense of the initial energy, -infty< E(0)<+infty.

GLOBAL NONEXISTENCE OF ARBITRARY INITIAL ENERGY SOLUTIONS OF VISCOELASTIC EQUATION WITH NONLOCAL BOUNDARY DAMPING

In this paper, we consider the long time behavior of solutions of the initial value problem for the viscoelastic wave equation under boundary damping \begin{eqnarray*} u_{tt} - \Delta u + \int_0^t g(t-\tau) \text{div}(a(x)\nabla u(\tau)) d\tau + u_t = 0 &\text{in}\, \Omega \times (0,\infty). \end{eqnarray*} For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the solution blows up in finite time.

GLOBAL NONEXISTENCE OF ARBITRARY INITIAL ENERGY SOLUTIONS OF VISCOELASTIC EQUATION WITH NONLOCAL BOUNDARY DAMPING

In this paper, we consider the long time behavior of solutions of the initial value problem for the viscoelastic wave equation under boundary damping \begin{eqnarray*} u_{tt}-\Delta u+\int_0^t g(t-\tau)\text{div}(a(x)\nabla u(\tau))d\tau+u_t=0 &\text{in}\,\Omega\times(0,\infty). \end{eqnarray*} For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the solution blows up in finite time.

On Global Solutions for the Cauchy Problem of a Boussinesq‐Type Equation

We will give conditions which will guarantee the existence of global weak solutions of the Boussinesq‐type equation with power‐type nonlinearity γ|u|p and supercritical initial energy. By defining new functionals and using potential well method, we readdressed the initial value problem of the Boussinesq‐type equation for the supercritical initial energy case.

Blowup of Smooth Solutions for Relativistic Euler Equations

We study the singularity formation of smooth solutions of the relativistic Euler equations in (3 + 1)-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any smooth solution, with compactly supported non-trivial initial data, blows up in finite time. For the case of infinite initial energy, we first prove the existence, uniqueness and stability of a smooth solution if the initial data is in the subluminal region away from the vacuum. By further assuming the initial data is a smooth compactly supported perturbation around a non-vacuum constant background, we prove the property of finite propagation speed of such a perturbation. The smooth solution is shown to blow up in finite time provided that the radial component of the initial ``generalized'' momentum is sufficiently large.