Global solutions and self-similar solutions for nonlinear beam equations

This paper concerns with the global classical solution to the Cauchy problem of the nonlinear double dispersive wave equation with strong damping uttuutt + � 2 u cut = d N X i=1 ∂ ∂xi σ i(uxi), where c and d are positive constants. By the contraction mapping principle and priori estimates, we prove that the Cauchy problem admits a unique global classical solution, and by the concavity method, we give the sufficient condi- tions on the blowup of the global solution of the Cauchy problem. Finally, as an application, an example is also given.

This paper is devoted to a system of nonlinear impulsive differential equations with three-point boundary conditions. The Green function is constructed and considered original problem is reduced to the equivalent impulsive integral equations. Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the problem and Schaefer’s fixed point theorem is used to prove the existence of a solution of the problem under consideration. We illustrate the application of the main results by two examples.

We examine the effect of thermal conduction on the low-angular momentum hot accretion flow (HAF) around non-rotating black holes accreting mass at very low rate. While doing so, we adopt the conductive heat flux in the saturated form, and solve the set of dynamical equations corresponding to a steady, axisymmetric, viscous, advective accretion flow using numerical methods. We study the dynamical and thermodynamical properties of accreting matter in terms of the input parameters, namely energy (ε0), angular momentum (ℓ0), viscosity parameter (α), and saturation constant (Φs) regulating the effect of thermal conduction. We find that Φs plays a pivotal role in deciding the transonic properties of the global accretion solutions. In general, when Φs is increased, the critical point (rc) is receded away from the black hole, and flow variables are altered particularly in the outer part of the disc. To quantify the physically acceptable range of Φs, we compare the global transonic solutions with the self-similar solutions, and observe that the maximum saturation constant ($\Phi ^{\rm max}_{\rm s}$) estimated from the global solutions exceeds the saturated thermal conduction limit (Φsc) derived from the self-similar formalism. Moreover, we calculate the correlation between α and $\Phi ^{\rm max}_{\rm s}$ and find ample disagreement between global solutions and self-similar solutions. Further, using the global flow variables, we compute the Bernoulli parameter (Be) which remains positive all throughout the disc, although flow becomes loosely unbound for higher Φs. Finally, we indicate the relevance of this work in the astrophysical context in explaining the possibility of mass-loss/outflows from the unbound disc.

Objective: In this work, the existence of a global solution for a fourth-order nonlinear elliptic equation with boundary conditions is studied. Method: Variational methods are applied to the proposed model, naturally using the Banach Fixed Point Contraction Theorem. Results and Discussion: The conclusion of the result obtained is that for the nonlinear elliptic equation of the fourth order they have equivalent norms in the conveniently defined spaces. On the other hand, the existence of a solution exists whenever Banach's fixed-point theorem can be applied. Implications of the research: Apply in the first university years the application of global solution for a fourth-order nonlinear elliptic equation in all university careers and specialties as part of the management of the tools of Information and Communication Technology (TIC). Originality/Value: This paper also provides an alternative approach to Analysis of global solution for a fourth-order nonlinear elliptic equation.

变系数非线性梁方程的周期解

We study the existence and the asymptotic behaviour of global solutions of the semilinear parabolic equation u(0) = ϧwhere a, b ∈ℝ, q > 1, p > 1. Forq=2p/(p+1) and ½ 1(p-1)>1 (equivalently, q > (n + 2)/(n + 1)), we prove the existence of mild global solutions for small initial data with respect to some norm. Some of those solutions are asymptotically self-similar.

Existence for continuous nonoscillatory solutions of second-order nonlinear difference equations with continuous variable

On the optimal control problem governed by the nonlinear elastic beam equation

In this article the existence as well as the uniqueness (EU) of the solutions for nonlinear multiorder fractional-differential equations (FDE) with local boundary conditions and fractional derivatives of different orders (Caputo and Riemann–Liouville) are covered. The existence result is derived from Krasnoselskii’s fixed point theorem and its uniqueness is shown using the Banach contraction mapping principle. To illustrate the reliability of the results, two examples are given.

In this paper, we will discuss some results on the existence and uniqueness of mild and strong solution of initial value problem of fractional order subjected to non-local conditions, by using the Banach fixed point theorem and the theory of strongly continuous cosine family under Caputo sense. Furthermore, we also prove that solution of Nonlinear Fractional Volterra Integrodifferential Equations and Nonlinear Fractional Mixed Integrodifferential Equations With Nonlocal Conditions is unique. Moreover, examples demonstrate the validity of the obtained main result and we obtain the solution for an equation, and proved that this solution is unique.

Well-posedness, global existence and large time behavior for Hardy–Hénon parabolic equations

A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations

This paper particularly addresses and discusses some analytical studies on the existence and uniqueness of global or blow-up solutions under the traveling profile forms for a free boundary problem of two-dimensional diffusion equations of moving fractional order. It does so by applying the properties of Schauder's and Banach's fixed point theorems. For application purposes, some examples of explicit solutions are provided to demonstrate the usefulness of our main results.

We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the $L^2(\mathbb{R}^d)$-framework.

A critical function of polymeric matrices in biological systems is to exert selective control over the transport of thousands of nanoparticulate species. Utilizing “third-party” molecular anchors to crosslink nanoparticulates to the matrix is an effective strategy, and a trapped nanoparticulate formed a desired complex MAP that is necessary to keep the nanoparticulate immobilized at any given time. In this paper, the global solution and stability of a parabolic–ordinary-parabolic haptotaxis system to complex MAP are studied. First, the existence of a local classical solution to system (4) has been observed using fixed point argument and parabolic Schauder estimates. Furthermore, some a priori estimates that can raise the regularity estimate of the solution for the relatively complicated first equation of system (3) from Lρ to L2ρ (ρ≥1) are given; then, the local classic solution can thus extend to the global classic solution when the space dimension N≤3. Lastly, by using various analytical methods, a threshold value ξ00(ξ00<0) is found, such that positive constant steady state (u∗,v∗,w∗) becomes unstable when ξ<ξ00. Our results show that the haptotaxis plays a crucial role in determining the stability to the model (3), that is, it can have a destabilizing effect.