The blow-up problem for exponential nonlinearities

Journal Article · Mon Jul 01 04:00:00 EDT 1996 · Communications in Partial Differential Equations

DOI:https://doi.org/10.1080/03605309608821177· OSTI ID:255086

Kichenassamy, S ^[1]

---

1. Univ. of Minnesota, Minneapolis, MN (United States)

We give a solution of the blow-up problem for equation {open_square}{upsilon}=e{sup {upsilon}}, with data close to constants, in any number of space dimensions: there exists a blow-up surface, near which the solution has ligarithmic behavior; its smoothness is estimated interms of the smoothness of the data. More precisely, we prove that for any solution of {open_square}{upsilon}=e{sup {upsilon}} with Cauchy data on t = 1 close to (ln2,-2) in H{sup s}(R{sup n}) x H{sup s-1}(R{sup n}), is a large enough integer, must blow-up on a space like hypersurface defined by an equation y={psi}({chi}) with {psi} {element_of} H{sup s}-146-9[n/2](R{sup n}). Furthermore, the solution has an asymptotic expansion ln(2/T{sup 2})+{Sigma}{sub j,k}{upsilon}{sub jk}(x)T{sup j+k}(lnT){sup k}, where T=t-{psi}{chi}, valid upto order s-151-10[n/T]. Logarithmic terms are absent if and only if the blow-up surface has vanishing scalar curvature. The blow-up time can be identified with the minimum of the function {psi}. Although attention is focused on one equation, the strategy is quite general; it consists in applying the Nash-Moser IFT to map from {open_quotes}singularity data{close_quotes} to Cauchy data.

• Sponsoring Organization: USDOE
• OSTI ID: 255086
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 1-2 Vol. 21; ISSN 0360-5302; ISSN CPDIDZ
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
CAUCHY PROBLEM
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
SINGULARITY