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

Honda, Hironobu

J. Math. Soc. Japan 56, No. 3, 907-922 (2004).

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.

Reviewer: José Bonet (Valencia)

MSC:

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

Keywords:

micro-local energy method

Citations:

Zbl 0891.35050