Decay rate of the range component of solutions to some semilinear evolution equations

We examine the rate of decay to 0, as t → +∞., of the projection on the range of A of the solutions of an equation of the form u′ + Au + |u| ^p−1 u = 0 or u′′ + u′ + Au + |u| ^p−1 u = 0 in a bounded domain of ^N , where A = −Δ with Neumann boundary conditions or A = −Δ − λ₁ I with Dirichlet boundary conditions. In general this decay is much faster than the decay of the projection on the kernel; it is often exponential, but apparently not always.     

Brake orbits type solutions to some class of semilinear elliptic equations

We consider a class of semilinear elliptic equations of the form

Long-time extinction of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equation of the following type t_∂u−Δu+a₀(x)uq=0 where , d₀>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.     

On the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearities

In this paper, we study the existence of positive solutions for the semilinear elliptic problem

     

Attractors for parabolic equations with dynamic boundary conditions

In this paper we prove existence of global solutions and (L²(Ω)×L²(Γ),(H¹(Ω)∩L^p(Ω))×L^p(Γ))$(L^2\left(\Omega \right)\times L^2\left(\Gamma \right),(H^1\left(\Omega \right)\cap L^p\left(\Omega \right))\times L^p\left(\Gamma \right))$-global attractors for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary, where there is no other restriction on p(≥2)$p(\ge 2)$.     

Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities

It is shown that a Lie point symmetry of the semilinear polyharmonic equations involving nonlinearities of power or exponential type is a variational/divergence symmetry if and only if the equation parameters assume critical values. The corresponding conservation laws for critical polyharmonic semilinear equations are established.     

Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory

We show the existence of monotone in time solutions for a semilinear parabolic equation with memory. The blow-up rate estimate of the solution is known to be a consequence of the monotonicity property.     

Phase space analysis of semilinear parabolic equations

We present a wave packet analysis of a class of possibly degenerate parabolic equations with variable coefficients. As a consequence, we prove local wellposedness of the corresponding Cauchy problem in spaces of low regularity, namely the modulation spaces, assuming a nonlinearity of analytic type. As another application, we deduce that the corresponding phase space flow decreases the global wave front set. We also consider the action on spaces of analytic functions, provided the coefficients are analytic themselves.     

Initial blow-up of solutions of semilinear parabolic inequalities

We study classical nonnegative solutions u(x,t) of the semilinear parabolic inequalities

     

On the non-existence of global solutions to the Cauchy problem for semilinear parabolic equations and inequalities

We generalize and improve recent non-existence results for global solutions to the Cauchy problem for the inequality as well as for the equation u_t = Δu + |u|^q in the half-space . Received: 16 September 2005     

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux-Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.     

Cauchy problems of semilinear pseudo-parabolic equations

This paper concerns with the Cauchy problems of semilinear pseudo-parabolic equations. After establishing the necessary existence, uniqueness and comparison principle for mild solutions, which are also classical ones provided that the initial data are appropriately smooth, we investigate large time behavior of solutions. It is shown that there still exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations.     

Blow-up for semilinear parabolic equations with nonlinear memory

In this paper, we consider the semilinear parabolic equation with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.     

On the analyticity of solutions to semilinear differential equations degenerated on a submanifold

We investigate the analyticity of solutions to semilinear elliptic equations degenerated on a submanifold. We introduce a new weighted Sobolev space which is appropriate for studying such equations. The technique for linear equations using cut-off functions cannot be applied and we need to use a representation formula which requires a fundamental solution.     

Heteroclinic connections for fully non-linear non-autonomous second-order differential equations

We investigate the solvability of the following strongly non-linear non-autonomous boundary value problem

     

A note on parabolic equation with nonlinear dynamical boundary condition

We consider a semilinear parabolic equation subject to a nonlinear dynamical boundary condition that is related to the so-called Wentzell boundary condition. First we prove the existence and uniqueness of global solutions as well as the existence of a global attractor. Then we derive a suitable ?ojasiewicz-Simon type inequality to show the convergence of global solutions to single steady states as time tends to infinity under the assumption that the nonlinear terms f,g are real analytic. Moreover, we provide an estimate for the convergence rate.     

Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with supercritical nonlinearity. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its sharp convergence rate for a class of initial data. We also derive a universal lower bound of the convergence rate which implies the optimality of the result. Proofs are given by a comparison method based on matched asymptotics expansion.     

Periodic orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z³+B(t)z²+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z³+B(t)z² studied in the literature, where the center variety is located in a finite number of connected components.