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The Cauchy problem for degenerate parabolic equations and Newton polygon. (English) Zbl 0909.35077
The author considers the Cauchy problem for a higher-order parabolic equation of the form \[ \partial_t^mu(t,x) -\sum_{j=1}^m\sum_{|\alpha|\leq pj}a_{j\alpha}(t,x) \partial^{\alpha}_x\partial^{m-j}_t u(t,x) =0,\quad 1\leq j\leq m, \] where \(t\in [0,T], p>0\). He investigates sufficient conditions for the Cauchy problem to be well-posed. He proves that under defined assumptions on the coefficients of this equation and the roots of a characteristic equation there exists \(T>0\) such that the Cauchy problem is \(H^{\infty}\)-wellposed. That is, for any \(\partial^{j-1}_tu(0,x) = \phi_j(x)\in H^{\infty}(\mathbb{R}^n),\) there exists a unique solution \(u(t,x)\in C^m_t([0,T];H^{\infty}(\mathbb{R}^n)).\) Moreover \(u(t,x)\in C^m_t((0,T];H^{\infty}_{1/p}(\mathbb{R}^n)),\) where \(H^{\infty}_{1/p}(\mathbb{R}^n)\) stands for the Gevrey class of exponent \(1/p\).

35K65 Degenerate parabolic equations
35K25 Higher-order parabolic equations