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The Cauchy problem for degenerate parabolic equations in Gevrey classes. (English) Zbl 0920.35067
The authors consider the Cauchy problem \[ P(t,x,\partial_t, D_x)u(t,x)= f(t,x),\quad (t,x)\in (0,T)\times \mathbb{R}^n, \] \[ \partial^j_t u(0,x)= u_j(x),\quad x\in \mathbb{R}^n,\quad j= 0,\dots,m-1, \] where \(P\) is a degenerate parabolic operator of the type \[ P= \partial^m_t+ \sum^m_{j=1} \sum_{|\alpha|\leq M} a_{j\alpha}(t,x) D^\alpha_x \partial^{m-j}_t \] with \(a_{j\alpha}(t,x)= t^{h(j,\alpha)} b_{j\alpha}(t,x)\). The exponents \(h(j,\alpha)\) are nonnegative integers. The Newton’s polygon determined by \(h(j,\alpha)\) is considered and, in terms of it, a definition of parabolicity for \(P\) is given. In this frame, the well-posedness of the Cauchy problem is proved in Gevrey classes of suitable order.
Reviewer: L.Rodino (Torino)
MSC:
35K30 Initial value problems for higher-order parabolic equations
35K65 Degenerate parabolic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators
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