Blow-up problems for nonlinear parabolic equations on locally finite graphs

Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation u_t=Δu + f(u) on G. The blow-up phenomenons for u_t=Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.     

Oblique derivative problems for second order nonlinear equations of mixed type in multiply connected domains

In this paper, the unique solvability of oblique derivative boundary value problems for second order nonlinear equations of mixed (elliptic-hyperbolic) type in multiply connected domains is proved, which mainly is based on the representation of solutions for the above boundary value problem, and the uniqueness and existence of solutions of the above problem for the equation u_xx + sgn y u_yy = 0.     

Existence of weak solutions of a nonlinear parabolic equation in a noncylindrical domain

Consider the mixed boundary value problem ?_tu + L[u] = f with a squareintegrable initial value and with zero boundary values in a domain Q. L[u] is a nonlinear elliptic operator in divergence form, defined on a domain with timedependent boundary. Weak solutions in cylindrical domains are used to construct a weak solution in Q by approximating Q by a system of cylinders. It is shown that this solution is continuously dependent on the initial value.     

Fourth order wave equations with nonlinear strain and source terms

In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u₀)?0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.     

Boundary value problems for the Fitzhugh-Nagumo equations

This paper is concerned with the study of the FitzHugh-Nagumo equations. These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon; they are a simplified version of the Hodgkin-Huxley equations. The FitzHugh-Nagumo equations consist of a non-linear diffusion equation coupled to an ordinary differential equation. v_t = v_xx + f(v) ? u, u_t = σv ? γu. We study these equations with either Dirichlet or Neumann boundary conditions, proving local and global existence, and uniqueness of the solutions. Furthermore, we obtain L_∞ estimates for the solutions in terms of the L₁ norm of the boundary data, when the boundary data vanish after a finite time and the initial data are zero. These estimates allow us to prove exponential decay of the solutions.     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Existence and non‐existence of global solutions for a class of non‐linear wave equations

This paper studies the existence and the non‐existence of global solutions to the initial boundary value problems for the non‐linear wave equation u_tt + u_xxxx = σ(u_x)_x + f(x, t) and the Boussinesq‐type equation u_tt + u_xxxx = σ(u)_xx + f(x, t). The paper proves that every above‐mentioned problem has a unique global solution under rather mild confining conditions, and arrives at some sufficient conditions of blow‐up of the solutions in finite time. Finally, a few examples are given. Copyright © 2000 John Wiley & Sons, Ltd.     

On the generalized Burgers equation

The present paper is concerned with the initial boundary value problem for the generalized Burgers equation u _t + g(t, u)u _x + f(t, u) = εu _xx which arises in many applications. We formulate a condition guaranteeing the a priori estimate of max |u _x | independent of ε and t and give an example demonstrating the optimality of this condition. Based on this estimate we prove the global existence of a unique classical solution of the problem and investigate the behavior of this solution for ε → 0 and t → + ∞. The Cauchy problem for this equation is considered as well.     

Nonlinear waves in friedman-robertson-walker space-times

In this paper we consider the initial value problem for the nonlinear wave equation □u = F(u, u′) in Friedman-Robertson-Walker space-time, □ being the D'Alambertian in local coordinates of space-time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.     

Spectral properties of the neutron transport equation

The general equation describing the steady-state flow through a porous column is λu ? D_xA(D_x?(u) + G(u)) = f, where λ is a nonnegative constant. In this paper existence, uniqueness and comparison results for solutions to the Dirichlet and mixed boundary value problems associated with this equation are proven. The existence of a weak solution to the evolution problems associated with the equation u_t = D_x(D_x?(u) + G(u)) are deduced.     

Blowup of solutions for the “bad” Boussinesq-type equation

The paper studies the blowup of solutions to the initial boundary value problem for the “bad” Boussinesq-type equation u_tt−u_xx−bu_xxxx=σ(u)_xx, where b>0 is a real number and σ(s) is a given nonlinear function. By virtue of the energy method and the Fourier transform method, respectively, it proves that under certain assumptions on σ(s) and initial data, the generalized solutions of the above-mentioned problem blow up in finite time. And a few examples are shown, especially for the “bad” Boussinesq equation, two examples of blowup of solutions are obtained numerically.     

Flows generated by symmetric functions of the Eigenvalues of the Hessian

Global unique solvability of the first initial boundary value problem for fully nonlinear equations of the form $$ - u_t + f(\lambda _1 [u], \ldots ,\lambda _n [u]) = g$$ is proved. Here λ _u , i=1,..., n, are the eigenvalues of the Hessian u_xx and f is a symmetric function satisfying some conditions. Bibliography: 7 titles.     

The finite-time blowup of the solution of an initial boundary-value problem for the nonlinear equation of ion sound waves

We obtain blowup conditions for the solutions of initial boundary-value problems for the nonlinear equation of ion sound waves in a hydrogen plasma in the approximation of “hot” electrons and “heavy” ions. A specific characteristic of this nonlinear equation is the noncoercive nonlinearity of the form ?_t|?u|², which complicates its study by any energy method. We solve this problem by the Mitidieri–Pohozaev method of nonlinear capacity.     

Solutions of multidimensional partial differential equations representable as a one-dimensional flow

We propose an algorithm for reducing an (M+1)-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow u_t + $w_{x_1 } $ (u, u_x u_xx,…) = 0 (where w is an arbitrary local function of u and its xi derivatives, i = 1,…, M) to a family of M-dimensional nonlinear PDEs F(u,w) = 0, where F is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the M-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original (M+1)-dimensional equation. Moreover, a spectral parameter can be introduced in the function F, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.     

On the continuation and boundedness of solutions of a nonlinear differential equation

Sufficient conditions are given so that all solutions of the nonlinear differential equation u″ + φ(t, u, u′)u′ + p(t) gf(u) g(u′) = h(t, u, u′) are continuable to the right of an initial t-value t₀ ? 0. These conditions are then extended so that all solutions u of the equation in question together with their derivative u′ are bounded for t ? t₀ .     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.     

Large time behavior of solutions to a degenerate parabolic equation not in divergence form

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: ut=upΔu+auq−bur, subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t→∞. In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.     

Initial boundary value problem for a class of fourth-order wave equation with viscous damping term

In this article we study the initial boundary value problem for a class of fourth-order nonlinear wave equation with viscous damping term u _tt ???αu _xxt ?+?u _xxxx ?=?f(u _x ) _x . By argument related to the potential well-convexity method, we prove the global existence and nonexistence of the solution. Further, we give some sharp conditions for global existence and nonexistence of the solution. This generalizes the results obtained in Chen and Lu [G. Chen and B. Lu, The initial-boundary value problems for a class of nonlinear wave equations with damping term, J. Math. Anal. Appl. 351 (2009), pp. 1–15].     

一类非线性高阶波动方程的初边值问题

该文研究一类非线性高阶波动方程u_tt－a_1^Uxx+a_₂u_x4+a₃u_x4_tt=φ(u_{x ⁾}x+f(u,u_x,u_xxu_xxx,u_x4)的初边值问题.证明整体古典解的存在唯一性并给出古典解爆破的充分条件.     

Initial boundary value problem for generalized boussinesque type equations with nonlinear source

A comparison principle for solutions of the first initial boundary value problem for the generalized Boussinesque equation with a nonlinear sourceu _t-Δψ(u)-Δu _t+q(u)=0 is established. By using this comparison principle, we prove new existence and nonexistence theorems for solutions of the first initial boundary value problem in the case of power-law functions ψ (ξ) andq (ξ). Translated fromMathematicheskie Zametki, Vol. 65, No. 1, pp. 70–75, January, 1999.