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.     

Nonlinear abstract differential equations with deviated argument

In this paper we consider the nonlinear differential equation with deviated argument u^′(t)=Au(t)+f(t,u(t),u[φ(u(t),t)]), t∈R₊, in a Banach space (X,‖⋅‖), where A is the infinitesimal generator of an analytic semigroup. Under suitable conditions on the functions f and φ, we prove a global existence and uniqueness result for the above equation.     

The existence of ground states for fourth-order wave equations

Focusing on the fourth-order wave equation u_tt+Δ²u+f(u)=0, we prove the existence of ground state solutions u=u(x+ct) for an optimal range of speeds c∈Rⁿ and a variety of nonlinearities f.     

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,∞[.     

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.     

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.     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.     

Positive and sign-changing solutions for fourth-order BVPs with parameters

This paper is concerned with the existence and multiplicity of positive and sign-changing solutions of the fourth-order boundary value problem u ⁽⁴⁾(t)=λ f(t,u(t),u ^′′(t)), 0<t<1,?u(0)?=?u(1)=u ^′′(0)=u ^′′(1)?=0, where f:[0,1]×?→? is continuous, λ∈? is a parameter. By using the fixed-point index theory of differential operators, it is proved that the above boundary value problem has positive, negative and sign-changing solutions for λ being different intervals. As an example, the boundary value problem u ⁽⁴⁾(t)+?η u ^′′(t)??ζu(t)=?λ f(t,u(t)), ?0<t<1,?u(0)=?u(1)=?u ^′′(0)=?u ^′′(1)=0 is also considered and some obtained results are the complement of the known results.     

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

We consider an Allen-Cahn type equation of the form ut=Δu+ε⁻²fε(x,t,u), where ε is a small parameter and fε(x,t,u)=f(u)−εgε(x,t,u) a bistable nonlinearity associated with a double-well potential whose well-depths can be slightly unbalanced. Given a rather general initial data u₀ that is independent of ε, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within the time scale of order ε²|lnε|, and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order ε. This is an optimal estimate that has not been known before for solutions with general initial data, even in the case where gε≡0.Next we consider systems of reaction-diffusion equations of the form

     

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.     

Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u⁽⁴⁾(t)-2u^″(t)+u(t)=f(u(t)),0<t<1, u^′(0)=u^′(1)=u^?(0)=u^?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory.     

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.     

A remark on Schrödinger operators

We study the almost everythere convergence to the initial dataf(x)=u(x, 0) of the solutionu(x, t) of the two-dimensional linear Schrödinger equation Δu=i? _t u. The main result is thatu(x, t) →f(x) almost everywhere fort → 0 iff ∈H ^p (R²), wherep may be chosen <1/2. To get this result (improving on Vega’s work, see [6]), we devise a strategy to capture certain cancellations, which we believe has other applications in related problems.     

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ =  λ* for the non-local porous medium equation u_t = (uⁿ)_xx + lf(u)/(ò_-1¹ f(u)dx)²{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s ^n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}.     

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.     

Blow-up results for a class of first-order nonlinear evolution inequalities

We investigate the non-existence of solutions to a class of evolution inequalities; in this case, as it happens in a relatively small number of blow-up studies, nonlinearities depend also on time-variable t and spatial derivatives of the unknown. The present results, which in great part do not require any assumption on the regularity of data, are completely new and shown with various applications. Some of these results referring to the problem u_t=Δu+a(x)|u|^p+λf(x) in R^N, t>0 include the non-existence results of positive global solutions obtained by Fujita and others when a≡1 and f≡0, Bandle-Levine and Levine-Meier when a≡|x|^m and f≡0, Pinsky when either f≡0 or f?0 and λ>0, Zhang and Bandle-Levine-Zhang when a≡1 and λ=1.     

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.     

Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control

In this work we provide an existence and location result for the third-order nonlinear differential equation

u^?(t)=f(t,u(t),u^′(t),u^″(t)),     

Global attractor for the Kirchhoff type equation with a strong dissipation

The paper studies the longtime behavior of the Kirchhoff type equation with a strong dissipation utt−M(‖∇u^‖2)Δu−Δut+h(ut)+g(u)=f(x). It proves that the related continuous semigroup S(t) possesses in the phase space with low regularity a global attractor which is connected. And an example is shown.