The parabolic Cauchy problem and quenching. (English) Zbl 1187.35120
Summary: This article treats the Cauchy problem for the equation
\[ u_t= \Delta u+ f(t,x,u) \]
With initial values \(u(0,x)= \varphi(x)\) in \(\mathbb R^N\) with an eye to the study of quenching phenomena. Among the results are an existence theorem under the sole assumption of continuity of \(f\), existence of maximal and minimal solutions without a monotonicity assumption regarding \(f\), an extension of the results on growth of solutions and on uniqueness that were obtained by Aguirre, Escobedo, and Herrero for power functions \(f(u)= u^p\) with \(0<p<1\) to a larger class of nonlinearities, using a new technique, and furthermore, new results on the behavior at infinity of the solution in just one fixed direction when the behavior of the initial function in this direction is known. The results are extended to parabolic systems, and some applications to quenching problems are discussed.
\[ u_t= \Delta u+ f(t,x,u) \]
With initial values \(u(0,x)= \varphi(x)\) in \(\mathbb R^N\) with an eye to the study of quenching phenomena. Among the results are an existence theorem under the sole assumption of continuity of \(f\), existence of maximal and minimal solutions without a monotonicity assumption regarding \(f\), an extension of the results on growth of solutions and on uniqueness that were obtained by Aguirre, Escobedo, and Herrero for power functions \(f(u)= u^p\) with \(0<p<1\) to a larger class of nonlinearities, using a new technique, and furthermore, new results on the behavior at infinity of the solution in just one fixed direction when the behavior of the initial function in this direction is known. The results are extended to parabolic systems, and some applications to quenching problems are discussed.
MSC:
| 35K15 | Initial value problems for second-order parabolic equations |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35K45 | Initial value problems for second-order parabolic systems |
