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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2007 • Question 3

Question

Math 6410 • ODE Fall 2007 • Question 3

Use an indirect argument (and basic existence uniqueness principles) to prove the following:

If \(u:\mathbb {R}\to \mathbb {R}\) belonging to \(C^2(-\infty ,\infty )\) is a one-signed solution (i.e. it either is everywhere nonnegative or everywhere nonpositve) of the equation \[ -u''+V(t)u=u^4, \] where \(V:\mathbb {R}\to \mathbb {R}\) is continuous, then \(u\) is either identically zero or never zero.

Raw LaTeX

Use an indirect argument (and basic existence uniqueness principles) to prove the
following:

If \(u:\R\to\R\) belonging to \(C^2(-\infty,\infty)\) is a one-signed solution
(i.e. it either is everywhere nonnegative or everywhere nonpositve) of the
equation
\[
  -u''+V(t)u=u^4,
\]
where \(V:\R\to\R\) is continuous, then \(u\) is either identically zero or never zero.

Topic Tags

Existence and Uniqueness Theorems Global Existence and Finite-Time Blow-Up Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions