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On a necessary condition for analytic well posedness of the Cauchy problem for parabolic equation. (English) Zbl 1061.35029

The author considers the Cauchy problem for the following parabolic equation with coefficients depending only on the variable \(x\): \[ \partial_t u(t,x)= a(x,D_x)u(t,x) + b(x,D_x) u(t,x), \;\;(t,x) \in [0,T]\times \mathbb R^s, \] with the initial condition \(u(0,x)=u_0(x), x \in \mathbb R^s\). Here \[ a(x,D_x)= \sum_{i,j} a_{i,j}(x)D_i D_ju, \quad b(x,D_x)=\sum_i b_i(x)D_iu + c(x)u, \] and the coefficients \(a_{i,j}(x), b_i(x), c(x)\) are real analytic functions on \(\mathbb R^s\). Inspired by the work of P. D’Ancona and S. Spagnolo [Math. Ann. 309, 307–330 (1997; Zbl 0891.35050)], the author defines when the Cauchy problem mentioned above is analytically well posed in \([0,T]\). He then proves that if the Cauchy problem is analytically well posed, then \(\operatorname{Re} a(x,\xi) \leq 0\) for all \((x,\xi) \in \mathbb R^s \times \mathbb R^s\). Moreover, if \(\operatorname{Re} a(x_0,\xi_0)=0\) at some point \((x_0,\xi_0)\), then \(\operatorname{Re} a(x,\xi_0)=0\) for every \(x \in \mathbb R^s\). The proof is long, proceeds by contradiction, and uses a micro-local energy method due to Mizohata.

MSC:

35K15 Initial value problems for second-order parabolic equations
35A20 Analyticity in context of PDEs

Citations:

Zbl 0891.35050
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