Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations

In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ²u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution.     

Finite-dimensional attractors for the Kirchhoff equation with a strong dissipation

The paper studies the existence of the finite-dimensional global attractors and exponential attractors for the dynamical system associated with the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the above mentioned dynamical system possesses a global attractor which has finite fractal dimension and an exponential attractor. For application, the fact shows that for the concerned viscoelastic flow the permanent regime (global attractor) can be observed when the excitation starts from any bounded set in phase space, and the dimension of the attractor, that is, the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow, is finite.     

An application of semigroups of locally Lipschitz operators to Carrier equations with acoustic boundary conditions

A generation theorem of semigroups of locally Lipschitz operators on a subset of a real Banach space is given and applied to the problem of the well-posedness of the Carrier equation utt−κ(‖u^‖2)Δu+γ|ut|^p−1ut=0 in Ω×(0,∞) with acoustic boundary condition, where p>2 and Ω is a bounded domain in an arbitrary dimensional space.     

Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as _∂ttu−Δu=f(x,u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as t_∂u−Δu=f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.     

Estimates on fractional power dissipative equations in function spaces

We study the fractional power dissipative equations, whose fundamental semigroup is given by e^{−t(−Δ)α} with α>0. By using an argument of duality and interpolation, we extend space-time estimates of the fractional power dissipative equations in Lebesgue spaces to the Hardy spaces and the modulation spaces. These results are substantial extensions of some known results. As applications, we study both local and global well-posedness of the Cauchy problem for the nonlinear fractional power dissipative equation u_t+(−Δ)^αu=|u|^mu for initial data in the modulation spaces.     

A nonlocal convection-diffusion equation

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut=J∗u−u+G∗(f(u))−f(u) in Rd, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equation ut=Δu+b⋅∇(f(u)). In fact, we prove that solutions of the nonlocal equation converge to the solution of the usual convection-diffusion equation when we rescale the convolution kernels J and G appropriately. Finally we study the asymptotic behaviour of solutions as t→∞ when f(u)=|u|^q−1u with q>1. We find the decay rate and the first-order term in the asymptotic regime.     

A non-autonomous strongly damped wave equation: Existence and continuity of the pullback attractor

In this paper we consider the strongly damped wave equation with time-dependent terms

u_tt−Δu−γ(t)Δu_t+β_ε(t)u_t=f(u),     

Formal solutions of semilinear heat equations

We study formal power series solutions to the initial value problem for semilinear heat equation t_∂u−Δu=f(u) with polynomial nonlinearity f and prove that they belong to the formal Gevrey class G². Next we give counterexamples showing that the solution, in general, is not analytic in time at t=0.     

Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ^∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ^∗, λ=λ^∗ and λ>λ^∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ‖=0 and limλ→+∞‖uλ‖=+∞.     

On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation

Consider the focusing mass-critical nonlinear Hartree equation iut+Δu=−(⁻²|⋅|∗²|u|)u for spherically symmetric initial data with ground state mass M(Q) in dimension d?5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling.     

Existence of positive periodic solutions of nonlinear first-order delayed differential equations

We consider the existence of positive ω-periodic solutions for the equation

u^′(t)=a(t)g(u(t))u(t)−λb(t)f(u(t−τ(t))),     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.     

Complete quenching for singular parabolic problems

We prove finite time extinction of the solution of the equation ut−Δu+χ{u>0}(u−β−λf(u))=0 in Ω×(0,∞) with boundary data u(x,t)=0 on ∂Ω×(0,∞) and initial condition u(x,0)=u₀(x) in Ω, where Ω⊂RN is a bounded smooth domain, 0<β<1 and λ>0 is a parameter. For every small enough λ>0 there exists a time t₀>0 such that the solution is identically equal to zero.     

The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function

The Nehari manifold for the equation −Δu(x)=λa(x)u(x)+b(x)|u(x)|^ν−1u(x) for x∈Ω together with Dirichlet boundary conditions is investigated. Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form t→J(tu) where J is the Euler functional associated with the equation) we discuss how the Nehari manifold changes as λ changes and show how existence and non-existence results for positive solutions of the equation are linked to properties of the manifold.     

Global attractor for a nonlinear wave equation arising in elastic waveguide model

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in an elastic waveguide model _utt−Δu−Δ_utt+Δ²u−Δ_ut−Δg(u)=f(x)$u_{tt}-\Delta u-\Delta u_{tt}+{\Delta }^{2}u-\Delta u_t-\Delta g\left(u\right)=f\left(x\right)$. It proves that when the space dimension N≤5$N\le 5$, under rather mild conditions the dynamical system associated with the above-mentioned IBVP possesses a global attractor which is connected and has finite fractal and Hausdorff dimension.     

Universal self-similarity of porous media equation with absorption: the critical exponent case

In this paper we study the large time behavior of non-negative solutions to the Cauchy problem of u_t=Δu^m−u^q in R^N×(0,∞), where m>1 and q=q_c≡m+2/N is a critical exponent. For non-negative initial value u(x,0)=u₀(x)∈L¹(R^N), we show that the solution converges, if u₀(x)(1+|x|)^k is bounded for some k>N, to a unique fundamental solution of u_t=Δu^m, independent of the initial value, with additional logarithmic anomalous decay exponent in time as t→∞.     

Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations t_∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m.     

Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation

∇⋅(uⁿ|∇Δu|^q−2∇Δu)−δu^mΔu=f(x,u)     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

Second critical exponent and life span for pseudo-parabolic equation

This paper studies the second critical exponent and life span of solutions for the pseudo-parabolic equation u_t−kΔu_t=Δu+u^p in Rⁿ×(0,T), with p>1, k>0. It is proved that the second critical exponent, i.e., the decay order of the initial data required by global solutions in the coexistence region of global and non-global solutions, is independent of the pseudo-parabolic parameter k. Nevertheless, it is revealed that the viscous term kΔu_t relaxes restrictions on the amplitude of the initial data required by the global solutions. Moreover, it is observed that the life span of the non-global solutions will be delayed by the third order viscous term. Finally, some numerical examples are given to illustrate all these results.