73 research outputs found
The nonlinear heat equation involving highly singular initial values and new blowup and life span results
In this paper we prove local existence of solutions to the nonlinear heat
equation with initial value , anti-symmetric with
respect to and
for where is a constant, and This gives a local
existence result with highly singular initial values.
As an application, for we establish new blowup criteria for
, including the case Moreover, if
we prove the existence of initial values
for which the resulting solution blows up in finite time
if is sufficiently small. We also construct blowing up solutions
with initial data such that has different finite
limits along different sequences . Our result extends the known
"small lambda" blow up results for new values of and a new class of
initial data.Comment: Submitte
Standing waves of the complex Ginzburg-Landau equation
We prove the existence of nontrivial standing wave solutions of the complex
Ginzburg-Landau equation with periodic boundary conditions. Our result includes all
values of and for which , but
requires that be sufficiently small
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
Self-Similar Subsolutions and Blowup for Nonlinear Parabolic Equations
AbstractFor a wide class of nonlinear parabolic equations of the formut−Δu=F(u,∇u), we prove the nonexistence of global solutions for large initial data. We also estimate the maximal existence time. To do so we use a method of comparison with suitable blowing up self-similar subsolutions. As a consequence, we improve several known results onut−Δu=up, on generalized Burgers' equations, and on other semilinear equations. This method can also apply to degenerate equations of porous medium type and provides a unified treatment for a large class of problems, both semilinear and quasilinear
A Fujita-type blowup result and low energy scattering for a nonlinear Schr\"o\-din\-ger equation
In this paper we consider the nonlinear Schr\"o\-din\-ger equation . We prove that if and
, then every nontrivial -solution blows up in finite or
infinite time. In the case and , we improve the existing low energy scattering results in dimensions . More precisely, we prove that if , then small data give rise to global, scattering
solutions in
Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state
Large time behavior of solutions to the nonlinear heat equation with absorption with highly singular antisymmetric initial values
In this paper we study global well-posedness and long time asymptotic
behavior of solutions to the nonlinear heat equation with absorption, , where and . We focus particularly on highly
singular initial values which are antisymmetric with respect to the variables
for some , such as , . In fact, we show global well-posedness
for initial data bounded in an appropriate sense by , for any .
Our approach is to study well-posedness and large time behavior on sectorial
domains of the form , and then to extend the results by reflection to solutions on which are antisymmetric. We show that the large time behavior depends on
the relationship between and , and we consider all three
cases, equal to, greater than, and less than . Our
results include, among others, new examples of self-similar and asymptotically
self-similar solutions
Singularity formation and blowup of complex-valued solutions of the modified KdV equation
The dynamics of the poles of the two--soliton solutions of the modified
Korteweg--de Vries equation are determined. A
consequence of this study is the existence of classes of smooth,
complex--valued solutions of this equation, defined for ,
exponentially decreasing to zero as , that blow up in finite
time
Finite-time blowup for a complex Ginzburg-Landau equation
We prove that negative energy solutions of the complex Ginzburg-Landau
equation blow up in finite time,
where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value , we
obtain estimates of the blow-up time as . It turns out that stays bounded (respectively, goes to
infinity) as in the case where the solution of the
limiting nonlinear Schr\"odinger equation blows up in finite time
(respectively, is global).Comment: 22 page
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