576 research outputs found
Modified scattering for the critical nonlinear Schr\"odinger equation
We consider the nonlinear Schr\"odinger equation in all dimensions , where and . We construct a class of initial values for which
the corresponding solution is global and decays as , like if and like if
. Moreover, we give an asymptotic expansion of those solutions
as . We construct solutions that do not vanish, so as to avoid
any issue related to the lack of regularity of the nonlinearity at . To
study the asymptotic behavior, we apply the pseudo-conformal transformation and
estimate the solutions by allowing a certain growth of the Sobolev norms which
depends on the order of regularity through a cascade of exponents
Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value
We consider the nonlinear heat equation on
, where and . We prove that in the range , there exist infinitely many
sign-changing, self-similar solutions to the Cauchy problem with initial value
. The construction is based on the
analysis of the related inverted profile equation. In particular, we construct
(sign-changing) self-similar solutions for positive initial values for which it
is known that there does not exist any local, nonnegative solution
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