Blow-up of solutions of semilinear parabolic differential equations. (English) Zbl 0661.35051
The Dirichlet initial boundary value problem for parabolic equations of type \(u_ t-Lu=h(x,t)f(u)\) (where \(L=a_{ij}(x,t)D_{ij}+b_ i(x,t)D_ i)\) is considered. Using comparison functions, we establish criteria on blow-up of the solution in finite time and give bounds for the blow-up time.
Applications in both bounded and unbounded domains are treated. The most significant example is the domain \({\mathbb{R}}^ n_ k=\{x|\) \(x_ i>0\), \(i=1,...,k\}\) (0\(\leq k\leq n)\), for which we also give sufficient conditions for the existence of global solutions. We obtain a generalization of Fujita’s well-known result for the case \(L=\Delta\), \(f(u)=u^ p\).
Applications in both bounded and unbounded domains are treated. The most significant example is the domain \({\mathbb{R}}^ n_ k=\{x|\) \(x_ i>0\), \(i=1,...,k\}\) (0\(\leq k\leq n)\), for which we also give sufficient conditions for the existence of global solutions. We obtain a generalization of Fujita’s well-known result for the case \(L=\Delta\), \(f(u)=u^ p\).
Reviewer: P.Meier
MSC:
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
semilinear; Dirichlet initial boundary value problem; comparison functions; blow-up; finite time; existence; global solutionsReferences:
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