Existence of solutions of nonlinear singular parabolic problems. (English) Zbl 0597.35070
The author considers the semilinear singular parabolic problem
\[
L_ ku=f(x,u)\quad on\quad \Omega:=(0,1)\times (0,T),
\]
where T is a positive number, f a regular function and, for k real:
\[
L_ ku:=u_{xx}+(k/x)u_ x-u_ t.
\]
The above parabolic equation is supplemented with initial conditions and boundary conditions as follows: let \(\Phi\) (x,t) be a given function: then it is required that \(u(x,0)=\Phi (x,0)\) on \(B_ 0:=(0,1)\times \{0\},\) and, denoting \(\{\) (r,t): \(0\leq t<T\}\) by \(S_ r\), either \(u(x,t)=\Phi (x,t)\) on \(S_ 1\cup S_ 0\) (problem I), or \(u(x,t)=\Phi (x,t)\) on \(S_ 1\) (problem II).
In case of problem I, the solution is sought in the space \[ C^ 2(\Omega)\cap C(S_ 0\cup S_ 1\cup B_ 0), \] while for problem II the solution should belong to \[ C^ 2(\Omega)\cap C(\Omega \cup S_ 1\cup B_ 0). \] The author proves, by a suitable approximation procedure, that problem II has a solution for any real k, while problem I has a solution whenever \(k<1\). The proofs make (non-trivial) use of comparison techniques, whereby the notion of barrier function plays a crucial role.
In case of problem I, the solution is sought in the space \[ C^ 2(\Omega)\cap C(S_ 0\cup S_ 1\cup B_ 0), \] while for problem II the solution should belong to \[ C^ 2(\Omega)\cap C(\Omega \cup S_ 1\cup B_ 0). \] The author proves, by a suitable approximation procedure, that problem II has a solution for any real k, while problem I has a solution whenever \(k<1\). The proofs make (non-trivial) use of comparison techniques, whereby the notion of barrier function plays a crucial role.
Reviewer: P.de Mottoni
MSC:
35K55 | Nonlinear parabolic equations |
35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |