Existence and uniqueness of entire solutions for a reaction-diffusion equation

We consider entire solutions of u_t=u_xx-f(u), i.e. solutions that exist for all (x,t)∈R², where f(0)=f(1)=0<f^′(0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f^′(1)>0, we show that such entire solution exists and is unique up to space-time translations. In the case f^′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space-time translation of the other. Yet all entire solutions in the second family only differ by a space-time translation.     

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.     

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.     

Potential well method for Cauchy problem of generalized double dispersion equations

In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)_xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u₀)?0 or I(u₀)<0 are proved.     

Existence and decay rates of solutions to the generalized Burgers equation

In this paper we study the generalized Burgers equation u_t+(u²/2)_x=f(t)u_xx, where f(t)>0 for t>0. We show the existence and uniqueness of classical solutions to the initial value problem of the generalized Burgers equation with rough initial data belonging to , as well it is obtained the decay rates of u in L^p norm are algebra order for p∈[1,∞[.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

Study of weak solutions for parabolic equations with nonstandard growth conditions

The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for ut=div((uσ+d₀)|∇u|^p(x,t)−2∇u)+f(x,t). Localization property of weak solutions is also discussed.     

Maximum principles for bounded solutions of the telegraph equation in space dimensions two and three and applications

A maximum principle is proved for the weak solutions of the telegraph equation in space dimension three u_tt−Δ_xu+cu_t+λu=f(t,x), when c>0, λ∈(0,c²/4] and (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation u_tt−Δ_xu+cu_t=F(t,x,u). Also, they can be employed in the study of almost periodic solutions of the forced sine-Gordon equation. A counterexample for the maximum principle in dimension four is given.     

Growth estimates for solutions to initial-boundary value problems in viscoelasticity

For the abstract Volterra integro-differential equation u_tt ? Nu + ∝_?∞^t K(t ? τ) u(τ) dτ = 0 in Hilbert space, with prescribed past history u(τ) = U(τ), ? ∞ < τ < 0, and associated initial data u(0) = f, u_t(0) = g, we establish conditions on K(t), ? ∞ < t < + ∞ which yield various growth estimates for solutions u(t), belonging to a certain uniformly bounded class, as well as lower bounds for the rate of decay of solutions. Our results are interpreted in terms of solutions to a class of initial-boundary value problems in isothermal linear viscoelasticity.     

Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws

This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u_t + f(u)_x = u_xx on [0, 1], with the boundary condition u(0, t) = u₋(t) → u₋, u(1, t) = u₊(t) → u₊, as t → +∞ and the initial data u(x,0) = u₀(x) satisfying u₀(0) = u₋(0), u₀(1) = u₊(1), where u± are given constants, u₋ ≠ u₊ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u₋ 〈 u₊, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u₋ > u₊, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |u₋−u₊| is small. Moreover, when u_±(t) ≡ u_±, t ≥ 0, exponential decay rates are both obtained.     

An initial- and boundary-value problem for a model equation for propagation of long waves

An initial- and boundary-value problem for a model equation for small-amplitude long waves is shown to be well-posed. The model has the form u_t + u_x + uu_x ? vu_xx ? α²u_xxt = 0, where x? [0, 1] and t ? 0. The solution u = u(x, t) is specified at t = 0 and on the two boundaries x = 0 and x = 1. Unique classical solutions are shown to exist, which depend continuously on variations of the specified data within appropriate function classes.     

Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation u_t = u_xx + f(u),   0 < u(x,t) < 1, (x,t) ∈ ℝ², where ƒ is a KPP‐type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling‐wave‐type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5‐dimensional, one is 4‐dimensional, and two are 3‐dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling‐wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Special exact solutions of the K(2, 2) equation, u_t + (u²)_x + (u²)_xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim_x→±∞u = A ≠ 0, or possesses compacton solutions only when lim_x→±∞u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Lipschitz metric for the periodic Camassa-Holm equation

We study the stability of conservative solutions of the Cauchy problem for the Camassa-Holm equation ut−uxxt+κux+3uux−2uxuxx−uuxxx=0 with periodic initial data u₀. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t),v(t))?eCtdD(u₀,v₀). The relationship between this metric and usual norms in and is clarified.     

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→∞.     

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 .     

Critical Exponent for the Parabolic Equation u_t=Δu~m + h(t)u~p in a Cone

In this paper,we study the initial-boundary value problem of porous medium equation ut = Δum + h(t)up in a cone D =(0,∞) ×Ω,where h(t) ～ tσ.Let ω1 denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l 2 +(n 2)l = ω1.We prove that if m p ≤ m + 2(σ+1) n+l + σ(m 1),then the problem has no global nonnegative solutions for any nonnegative u0 unless u0 = 0;if p m + 2(σ+1) n+l + σ(m1),then the problem has global solutions for some u 0 ≥ 0.