Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane

Details Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane

2002-03-05

arxiv.org/abs/math/0203045v1

Communications on Pure and Applied Mathematics, Vol. LIII, 0001- 0026 (2000))

Mathematics

Analysis of PDEs

Scientific paper

We prove existence and uniqueness results for nonlinear third order partial differential equations of the form $$ f_t - f_{yyy} = \sum_{j=0}^3 b_j (y, t; f) ~f^{(j)} + r(y, t) $$ where superscript $j$ denotes the $j$-th partial derivative with respect to $y$. The inhomogeneous term $r$, the coefficients $b_j$ and the initial condition $f(y,0)$ are required to vanish algebraically for large $|y|$ in a wide enough sector in the complex $y$-plane. Using methods related to Borel summation, a unique solution is shown to exist that is analytic in $y$ for all large $|y|$ in a sector. Three partial differential equations arising in the context of Hele-Shaw fingering and dendritic crystal growth are shown to be of this form after appropriate transformation, and then precise results are obtained for them. The implications of the rigorous analysis on some similarity solutions, formerly hypothesized in two of these examples, are examined.

Affiliated with

Also associated with

No associations

Rating

Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-125316