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Remarks on formal solution and genuine solutions for some nonlinear partial differential equations. (English) Zbl 1276.35054

Summary: S. Ōuchi ([J. Math. Sci., Tokyo 1, No. 1, 205–237 (1994; Zbl 0810.35006); J. Math. Sci., Tokyo 2, No. 2, 375–417 (1995; Zbl 0860.35018)]) found a formal solution \(\widetilde u(t,x)=\sum_{k\geq 0}u_k(x)t^k\) with \[ | u_k(x)|\leq AB^k\Gamma\biggl(\frac{k}{\gamma_\ast}+1\biggl)\quad 0<\gamma_\ast\leq\infty \] for some class of nonlinear partial differential equations. For these equations he showed that there exists a genuine solution \(u_S(t,x)\) on a sector \(S\) with asymptotic expansion \(u_S(t,x)\sim \widetilde u(t,x)\) as \(t\rightarrow 0\) in the sector \(S\). These equations have polynomial type nonlinear terms.
In this paper we study a similar class of equations with the following nonlinear terms \[ \sum\limits_{| q|\geq 1}t^{\sigma_q}c_q(t,x)\prod\limits_{j+|\alpha|\leq m}\biggl\{\biggl(t\frac{\partial}{\partial t}\biggl)^j\biggl(\frac{\partial}{\partial x}\biggl)^\alpha u(t,x)\biggl\}^{q_{j,\alpha}}. \] It is main purpose to get a solvability of the equation in a category \(u_S(t,x)\sim 0\) as \(t\rightarrow 0\) in a sector \(S\). We give a proof by the method that is a little different from that in [loc. cit.]. Further we give a remark that the similar class of equations has a genuine solution \(u_S(t,x)\) with \(u_S(t,x)\sim\widetilde u(t,x)\) as \(t\rightarrow 0\) in the sector \(S\).

MSC:

35C10 Series solutions to PDEs
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References:

[1] Gérard, R. and Tahara, H., Singular Nonlinear Partial Differential Equations , Vieweg, 1996. · Zbl 0874.35001
[2] Ōuchi, S., Formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo 1 (1994), 205-237. · Zbl 0810.35006
[3] Ōuchi, S., Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo 2 (1995), 375-417. · Zbl 0860.35018
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