Enclosure statements for systems of semilinear parabolic differential equations. (Einschliessungsaussagen bei Systemen semilinearer parabolischer Differentialgleichungen.) (German) Zbl 0741.35028

The author considers semilinear parabolic differential equations of the form \[ Mu(x,t)=u_ t(x,t)+{\mathcal L}[u](x,t)+f(x,t,u(x,t),u_ x(x,t)),\quad ((x,t)\in \Omega_ 0\times (0,t]), \] where \({\mathcal L}\) is a second order linear differential operator containing only derivatives with respect to \(x\in \Omega_ 0\subset \mathbb{R}^ m\). If \(v\) is a solution of \(M v=0\) satisfying certain initial and boundary conditions, estimates of the range of \(v\) of the form \[ v(x,t)\in\psi(x,t)G,\quad ((x,t)\in G\times [0,T]) \] are derived where \(G\) is a suitable subset of \(\mathbb{R}^ n\).
In case \(f\) only depends on \(u\), and the Dirichlet boundary condition \(u(x,t)=u_ s\) holds for \((x,t)\in \partial\Omega_ 0\times (0,\infty)\), \(u_ s\) is called a stationary point with respect to \(f\) if \(f(u_ s)=0\). In this case, stability and asymptotic stability are investigated. This means, if \(u(\cdot,0)\) is close to \(u_ s\), remains \(u(\cdot,t)\) close to \(u_ s\) for all \(t\geq 0\), or does \(u(\cdot,t)\to u_ s\) holds as \(t\to \infty\), respectively?
The general results are applied to a model of a biochemical reaction and a model for the nerve membrane.


35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B35 Stability in context of PDEs
92C20 Neural biology
92C40 Biochemistry, molecular biology
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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