Global Estimates and Regularity of Retarded Parabolic Equations with Fast-growing Nonlinearities

Compactly Generated Shape Index Theory and its Application to a Retarded Nonautonomous Parabolic Equation

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, and define the H-shape cohomology index to develop the Morse equations. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\sm E$ need not to be a neighborhood of the compact invariant set. Moreover, in this new theory, the phase space is not required to be separable. We apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to obtain the existence …

$S^1$-等变抛物方程全局吸引子连接图的分类

Classification of connection graphs of global attractors for $S^1$-equivariant parabolic equations

针对$S^1$上带可逆非线性项的一维半线性抛物方程，证明同lap signature的可逆非线性项间存在保持异宿连接的可逆同伦，完成其全局吸引子连接图的分类，并给出相关分岔图。

Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of an isolated invariant set. As the frequency of the oscillations tends to infinity, we prove that every isolated invariant set of the averaged autonomous equation can be continued to an isolated invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index …

On a class of degenerate parabolic equations with dynamic boundary conditions

We consider a quasi-linear parabolic equation with nonlinear dynamic boundary conditions occurring as a natural generalization of the semilinear reaction-diffusion equation with dynamic boundary conditions. The corresponding class of initial and boundary value problems has already been studied previously, proving well-posedness and the existence of the global attractor. The goal of this note is to show that the previous analysis can be redone for more general nonlinearities by proving an additional (uniform) L\infty-estimate on the solutions. In particular, we derive new conditions which refle…

Attractors for the semiflow associated with a class of doubly nonlinear parabolic equations

A doubly nonlinear parabolic equation of the form $α(u_t)-Δu+W'(u)= f$, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal monotone function $α$ and by the derivative $W'$ of a smooth but possibly nonconvex potential $W$; $f$ is a known source. After defining a proper notion of solution and recalling a related existence result, we show that from any initial datum emanates at least one solution which gains further regularity for $t>0$. Such "regularizing solutions" constitute a semiflow $S$ for…

Uniform Decay Estimates for Solutions of a Class of Retarded Integral Inequalities

Some uniform decay estimates are established for solutions of the following type of retarded integral inequalities: $$y(t)\leq E(t,τ)||y_τ||+\int_τ^t K_1(t,s)||y_s||ds+\int_t^\infty K_2(t,s)||y_s||ds+ρ, \hspace{0.5cm} t\geqτ\geq 0.$$ As a simple example of application, the retarded scalar functional differential equation $\dot x=-a(t)x+B(t,x_t)$ is considered, and the global asymptotic stability of the equation is proved under weaker conditions. Another example is the ODE system $\dot x=F_0(t,x)+\sum_{i=1}^m F_i(t,x(t-r_i(t)))$ on $R^n$ with superlinear nonlinearities $F_i$ ($0\leq i\leq m$). …