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Global solutions of nonlinear parabolic equations. (English) Zbl 0801.35044

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1992, No.XIX, 7 p. (1992).
From the text: We study the following Cauchy problem \[ u_ t= \Delta u+ \mu|\nabla u|\quad\text{ in }\mathbb{R}^ n\times\overline\mathbb{R}_ +,\quad u({\mathbf x},0)= u_ i({\mathbf x}),\;{\mathbf x}\in \mathbb{R}^ n.\tag{1} \] In (1) we are assuming that \(\mu\neq 0\) is a real constant. The main result is
Theorem: Let \(u_ 0\in C^ 3_ 0(\mathbb{R}^ n)\). Then there exists a unique classical solution to (1). Furthermore, this solution satisfies the maximum – minimum principle, \[ \sup_{\mathbb{R}^ n\times [0,\infty)} u({\mathbf x},t)= \sup_{\mathbb{R}^ n} u_ 0({\mathbf x}),\;\inf_{\mathbb{R}^ n\times [0,\infty)} u({\mathbf x},t)= \inf_{\mathbb{R}^ n} u_ 0({\mathbf x}). \]

MSC:

35K55 Nonlinear parabolic equations
35Q30 Navier-Stokes equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B50 Maximum principles in context of PDEs
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