Stability of stationary solutions of a semilinear parabolic partial differential equation

We consider a parabolic partial differential equation u_t = u_xx + f(u) on a compact interval of spatial variable x with Dirichlet boundary conditions. The stability of stationary solutions of this system is studied by the use of Liapunov's second method. We obtain necessary and sufficient conditions for the stability, asymptotic stability, neutral stability, instability, and conditional stability. These conditions are closely connected with the conditions for the existence of the stationary solutions.     

Blowup of solutions of a Korteweg-de Vries-type equation

We investigate the nonlinear third-order differential equation (u_xx ? u)_t + u _xxx + uu_x = 0 describing the processes in semiconductors with a strong spatial dispersion. We study the problem of the existence of global solutions and obtain sufficient conditions for the absence of global solutions for some initial boundary value problems corresponding to this equation. We consider examples of solution blowup for initial boundary value and Cauchy problems. We use the Mitidieri-Pokhozhaev nonlinear capacity method.     

Separation of variables of a generalized porous medium equation with nonlinear source

This paper considers a general form of the porous medium equation with nonlinear source term: u_t=(D(u)u_xⁿ)_x+F(u), n≠1. The functional separation of variables of this equation is studied by using the generalized conditional symmetry approach. We obtain a complete list of canonical forms for such equations which admit the functional separable solutions. As a consequence, some exact solutions to the resulting equations are constructed, and their behavior are also investigated.     

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.     

New solutions for the modified generalized Degasperis-Procesi equation

In this paper, using three distinct computational methods we obtain some new exact solutions for the generalized modified Degasperis-Procesi equation (mDP equation) u_t-u_xxt+(b+1)u²u_x=bu_xu_xx+uu_xxx. We show the graph of some of the new solutions obtained here with the aim to illustrate their physical relevance. Mathematica is used. Finally some conclusions are given.     

An elementary proof of the blow-up for semilinear wave equation in high space dimensions

This paper concerns the blow-up of solutions to u_tt−Δu=|u|^p in high dimensions for n?4 and 1<p<p₀(n), where p₀(n) is a critical exponent. We proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities.     

Plane structures in thermal runaway

We consider the problem

1. u _t=u _xx+e ^u whenx ∈ ?,t > 0,
2. u(x, 0) =u ₀(x) whenx ∈ ?,

whereu ₀(x) is continuous, nonnegative and bounded. Equation (1) appears as a limit case in the analysis of combustion of a one-dimensional solid fuel. It is known that solutions of (1), (2) blow-up in a finite timeT, a phenomenon often referred to as thermal runaway. In this paper we prove the existence of blow-up profiles which are flatter than those previously observed. We also derive the asymptotic profile ofu(x, T) near its blow-up points, which are shown to be isolated.     

Locally Lipschitz solutions of a single conservation law in several space variables

We consider the Cauchy problem for a single conservation law in several space variables. Letting u(x, t) denote the solution with initial data u₀, we state necessary and sufficient conditions on u₀ so that u(x, t) is locally Lipschitz continuous in the half space {t > 0}. These conditions allow for the preservation of smoothness of u₀ as well as for the smooth resolution of discontinuities in u₀. One consequence of our result is that u(x, t) cannot be locally Lipschitz unless u₀ has locally bounded variation. Another is that solutions which are bounded and locally Lipschitz continuous in {t > 0} automatically have boundary values u₀ at t = 0 in the sense that u(·, t) → u₀ in L_loc¹. Finally, we give an elementary proof that locally Lipschitz solutions satisfy Kruzkov's uniqueness condition.     

Globally smooth solutions of quasilinear hyperbolic systems in diagonal form

We prove the existence of a large class of globally smooth solutions of the Cauchy problem for the system of n equations u_t + Λ(x, t, u)u_x = 0, where Λ is a diagonal matrix. We show that, under certain monotonicity conditions on both Λ and the initial data u₀, the solution u will be locally Lipschitz continuous at positive times. Since u₀ is a function of locally bounded variation, our result thus provides both for the smoothing of discontinuities in u₀ as well as for the global preservation of smoothness. The global existence results from an a priori estimate of $\text{?u}\text{?x}$, which we obtain by a device which enables us to effectively uncouple the system of equations for $\text{?u}\text{?x}$. Finally, we prove a partial converse which demonstrates that our hypotheses are not overly restrictive.     

Discrete Toda lattices and the Laplace method

We apply the Laplace cascade method to systems of discrete equations of the form u _i+1,j+1 = f(u _i+1,j , u _i,j+1, u _i,j , u _i,j?1), where u _ij , i, j ∈ ?, is an element of a sequence of unknown vectors. We introduce the concept of a generalized Laplace invariant and the related property that the systems is “of the Liouville type.” We prove a series of statements about the correctness of the definition of the generalized invariant and its applicability for seeking solutions and integrals of the system. We give some examples of systems of the Liouville type.     

On existence of weak solutions for one-dimensional forward-backward diffusion equations

We establish the existence of infinitely many weak solutions for the general one-dimensional forward-backward diffusion equation u_t=σ(u_x)_x under the homogeneous Neumann boundary condition by rephrasing it as a first-order differential inclusion problem.     

Existence results for a class of second order evolution inclusions and its corresponding first order evolution inclusions

We deal with an abstract second order nonlinear evolution inclusion with its principal part having a small parameter ?. We prove the existence of a weak solution when the nonlinearity F is convex as well as nonconvex valued. Then we study the asymptotic behavior of a sequence of solutions {u _? } when ? → 0. We prove that there exists a limit function u, and u is a solution of the corresponding first order evolution inclusion.     

Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order

This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|^p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u₀. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u₀ as p→∞.     

Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in RN

In this paper, we study the existence of least energy sign-changing solutions for a Kirchhoff-type problem involving the fractional Laplacian operator. By using the constraint variation method and quantitative deformation lemma, we obtain a least energy nodal solution u_b for the given problem. Moreover, we show that the energy of u_b is strictly larger than twice the ground state energy. We also give a convergence property of u_b as b ↘ 0, where b is regarded as a positive parameter.     

Uniqueness of solutions to an Abel type nonlinear integral equation on the half line

We consider a convolution-type integral equation u = k ? g(u) on the half line (???; a), a ?? ?, with kernel k(x) = x ^???1, 0 < ??, and function g(u), continuous and nondecreasing, such that g(0) = 0 and 0 < g(u) for 0 < u. We concentrate on the uniqueness problem for this equation, and we prove that if ?? ?? (1, 4), then for any two nontrivial solutions u ₁, u ₂ there exists a constant c ?? ? such that u ₂(x) = u ₁(x +c), ??? < x. The results are obtained by applying Hilbert projective metrics.     

Nonexistence of solutions of the dirichlet problem for some quasilinear elliptic equations in a half-space

We prove the monotonicity of nonnegative bounded solutions of the Dirichlet problem for the quasilinear elliptic equation ?Δ_pu = f(u), p ≥ 3, in a half-space. This assertion implies new results on the nonexistence of solutions for the case in which f(u) = u^q with appropriate values of q.     

Spatially nonlocal problems with integral conditions for third-order differential equations

We obtain sufficient conditions for the existence of regular solutions of some nonlocal problems for the equation u _ttt + u _xx + μu = f(x, t) with conditions containing integrals with respect to the spatial variable.     

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.     

Fully nonlinear singularly perturbed equations and asymptotic free boundaries

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζε(u) with ζε→δ₀⋅∫ζ. We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F(D²u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.     

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 .