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A priori bounds, nodal equilibria and connecting orbits in indefinite superlinear parabolic problems. (English) Zbl 1143.37049

It is considered the parabolic problem \(u_t-\Delta u=\lambda u+a(x)g(u)+h(x,u)\), \(x\in\Omega\), \(t>0\); \(u=0\), \(x\in\partial\Omega\), \(t>0\), \(u(x,0)=u_0(x)\) on a smooth bounded domain \(\Omega\in\mathbb{R}^n\), \(a(x)\in L^\infty(\Omega)\), \(\lambda\in\mathbb{R}\), \(g\in C^1(\mathbb{R},\mathbb{R})\), \(g(0)=0\) is superlinear and subcritical, \(h:\Omega\times\mathbb{R}\to\mathbb{R}\), \(h(\cdot,0)=0\), has at most linear growth in \(u\in\mathbb{R}\), \(| u| \to\infty\). New a priori bounds for global semiorbits are obtained, that allow to derive known and new existence results for equilibria and prove the existence of connecting orbits. Also much information on nodal properties of equilibria is obtained.

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35K20 Initial-boundary value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems
47H20 Semigroups of nonlinear operators
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