Explosion in a diffusive strip due to a concentrated nonlinear source
The semilinear heat equation \(v_t= v_{xx}+ F(x, v)\) for \(0< x< \ell\), \(t> 0\), is studied for the temperature \(v(x, t)\), where the nonlinear source has a strongly concentrated and localized spatial dependence of the form \(F(x, v)= \delta(x- a) g(v)\) for \(0< a< \ell\), with \(\delta(x- a)\) being the Dirac delta distribution. The initial condition \(v(x, 0)= v_0(x)\), is imposed, and homogeneous Dirichlet or Neumann boundary conditions. It is assumed that \(g(v)\) is a smooth function and has the properties \(g(v)> 0\), \(g'(v)> 0\), \(g''(v)> 0\) for \(v> 0\), and \(\int^\infty_{h_0} dv/g(v)< \infty\) for some \(h_0> 0\). The authors prove that blow-up always occurs for the Neumann problem, whereas for the Dirichlet problem it depends upon the length \(\ell\) of the strip and proximity of the source site \(a\) to the boundary. For those situations in which blow-up does occur the growth rate is determined for certain class of nonlinearities. The proofs are achieved by reducing the initial-boundary value problem to a nonlinear Volterra equation for \(v(a, t)\), and then studying it for blow-up solutions and the growth rate of such blow-up.
scientific article; zbMATH DE number 818499