Asymptotic behaviour of solutions to semi-linear elliptic equations on the half-cylinder
The authors consider a non-prismatic cylinder of semi-infinite length with variable cross-section and plane base, \(\Omega=\{x\in\mathbb{R}^ 3:(x_ 2,x_ 2)\in D(x_ 3)\), \(x_ 3\in\mathbb{R}+\}\), where the bounded, plane cross-section \(D(x_ 3)\) has smooth boundary, and area \(| D(x_ 3)|\). On \(\Omega\), \(u\) satisfies the semi-linear equation \((a_{ij}u,_ i),_ j=f(u)\), with \(a_{ij}\) symmetric and uniformly elliptic. Further, \[ \lim_{x_ 3\to\infty}\int^{x_ 3}_ 0A^{- \mu}(\nu)d\nu=\infty,\quad\mu=2p/(1+p), \] where \(A(x_ 3)=| D(x_ 3)|^{(p-1)/2p}a_ 1^{1/2}(x_ 3)\), \(p>1\), and \(a_ 1(x_ 3)=\max_{X_ \alpha\in D(x_ 3)}a_{33}(x_ \alpha,x_ 3)\). The boundary conditions are either Dirichlet or Neumann. The main result of the paper is the following. If \(f\) satisfies \(uf(u)\geq c| u|^{2p}\), \(x_ 3>0\), then a smooth solution either fails to exist globally, or else it must tend asymptotically to zero as \(x_ 3\to\infty\). Various extensions are also discussed, including alternate conditions on the nonlinearity, and elliptic systems.
homogeneous Dirichlet boundary conditions
derivation of first order nonlinear differential inequality
scientific article; zbMATH DE number 63807