Global solvability for semilinear anisotropic evolution partial differential equations

✍ Scribed by Paola Marcolongo; Alessandro Oliaro

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 212 KB
• Volume: 47
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/s0362-546x(01)00197-3

✦ Synopsis

In this paper we consider the Cauchy problem for a class of semilinear anisotropic evolution equations with parabolic linear part. Using standard techniques we reduce our problem in an integral form. Thus a local (L^{2}) solution is given as fixed point of the correspondent integral operator, defined for (t \in\left[0, T_{1}\right]). Taking as initial datum the solution evalued in (T_{1}), we find a local (L^{2}) solution, defined for (t \in\left[T_{1}, 2 T_{1}\right]). Iterating this process and patching together all the local solutions defined on intervals of type (\left[n T_{1},(n+1) T_{1}\right]), with (n \in \mathbb{N}), we obtain a global solution defined for every (t \geq 0). We point out that our paper recaptures the results in Tadmor [8] as a particular case.