Infinite-dimensional singularity theory and bifurcation from equilibria for certain nonlinear evolution equations
The authors consider the semilinear parabolic problem \[ u_t- Lu- f(x, u)= g(x, t), \quad x\in \Omega, \qquad u=0, \quad x\in \partial \Omega. \tag{1} \] In (1), \(\Omega \subset \mathbb{R}^n\) is a bounded domain with smooth boundary \(\partial \Omega\), \(h\) is a uniformly elliptic formally selfadjoint second order differential operator with smooth coefficients, \(g\) is smooth and \(T\)-periodic with respect to \(t\), \(f\) is smooth, \(f_u\) is bounded, \(f(x, \cdot)\) is convex, and \[ 0< \lim_{s\to -\infty} {{f(x, s)} \over 2}< \lambda_1< \lim_{s\to \infty} {{f(x, s)} \over s}< \lambda_2, \] where \(\lambda_1\) and \(\lambda_2\) are the lowest two eigenvalues of \(-L\) with homogeneous Dirichlet boundary conditions. Let \(A: W_0^{1,2} (\Omega)\to W_0^{1,2} (\Omega)\) and \[ B: W^{1,2} ((0, T); L^2 (\Omega))\cap L^2 ((0, T); W_0^{1, 2} (\Omega))\to L^2 ((0, T); W_0^{1,2} (\Omega)) \] be defined by \[ \begin{aligned} \langle A(u), \varphi \rangle &:= \int_\Omega [uL\varphi+ f(x, u)\varphi ]dx,\\ \langle \langle B(u), \varphi \rangle \rangle &:= \int_0^T \int_\Omega [u_t \varphi- uL \varphi- f(x, u)\varphi ]dx dt, \end{aligned} \] where \(\langle \cdot, \cdot \rangle\) and \(\langle \langle \cdot, \cdot \rangle \rangle\) are the usual scalar product in \(W_0^{1, 2} (\Omega)\) and \(L^2 ((0, T); W_0^{1, 2} (\Omega))\), respectively. Then the main results may be formulated as follows: \(B\) is a \(C^1\)-Fredholm map of index zero. If (for all \(t\)), \(g(\cdot, t)\) belongs to a small neighbourhood of a singular value of \(A\) then (1) has exactly 2, 1 or 0 smooth \(T\)-periodic solutions in an appropriate neighbourhood of the associated singular point of \(A\).
scientific article; zbMATH DE number 828287