Search

This paper deals with the Landesman-Lazer type problem of elliptic equations associated with homogeneous Dirichlet boundary conditions. By using some dynamical arguments we derive some new results on bifurcation from infinity and multiplicity of the problems., Comment: 19 pages

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 of bounded full solutions.

We establish new global bifurcation theorems for dynamical systems in terms of local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation $u_t+A u=f_\lambda(u)$ in a Banach space $X$, where $A$ is a sectorial operator with compact resolvent. Assume that $0$ is always a trivial stationary solution of the equation. We show that the global dynamic bifurcation branch $\Gamma$ of a bifurcation point $(0,\lambda_0)$ either meets another bifurcation point $(0,\lambda_1)$, or is unbounded, completely extending the well-known Rabinowitz Global Bifurcation Theorem on operator equations to nonlinear evolution equations without any restrictions on the crossing number. In the case where $f_\lambda(u)=\lambda u+f(u)$, due to the {\em nonnegativity} of the Conley index we can even prove a stronger conclusion asserting that only one possibility occurs for $\Gamma$, that is, $\Gamma$ is necessarily unbounded. This result can be expected to help us have a deeper understanding of the dynamics of nonlinear evolution equations. As another example of applications of the abstract bifurcation theorems, we also discuss the bifurcation and the existence of nontrivial solutions of the elliptic equation $-\Delta u=f_\lambda(u)$ on a bounded domain in $\mathbb{R}^n$ ($n\geq 3$) associated with the homogenous Dirichlet boundary condition. Some new results with global features are obtained., Comment: 41pages, 6 figures

We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form $u_t+A u=f_\lambda(u)$ on a Banach space $X$, where $A$ is a sectorial operator, and $\lambda\in R$ is the bifurcation parameter. Suppose the equation has a trivial solution branch $\{(0,\lambda):\,\,\lambda\in R\}$. Denote $\Phi_\lambda$ the local semiflow generated by the initial value problem of the equation. It is shown that if the crossing number $n$ at a bifurcation value $\lambda=\lambda_0$ is nonzero and moreover, $S_0=\{0\}$ is an isolated invariant set of $\Phi_{\lambda_0}$, then either there is a one-sided neighborhood $I_1$ of $\lambda_0$ such that $\Phi_\lambda$ bifurcates a topological sphere $\mathbb{S}^{n-1}$ for each $\lambda\in I_1\setminus\{\lambda_0\}$, or there is a two-sided neighborhood $I_2$ of $\lambda_0$ such that the system $\Phi_\lambda$ bifurcates from the trivial solution an isolated nonempty compact invariant set $K_\lambda$ with $0\not\in K_\lambda$ for each $\lambda\in I_2\setminus\{\lambda_0\}$. We also prove that the bifurcating invariant set has nontrivial Conley index. Building upon this fact we establish a global dynamical bifurcation theorem. Roughly speaking, we prove that for any given neighborhood $\Omega$ of the bifurcation point $(0,\lambda_0)$, the connected bifurcation branch $\Gamma$ from $(0,\lambda_0)$ either meets the boundary $\partial\Omega$ of $\Omega$, or meets another bifurcation point $(0,\lambda_1)$. This result extends the well-known Rabinowitz's Global Bifurcation Theorem to the setting of dynamic bifurcations of evolution equations without requiring the crossing number to be odd. As an illustration example, we consider the well-known Cahn-Hilliard equation. Some global features on dynamical bifurcations of the equation are discussed., Comment: 42 pages