Decay and local eventual positivity for biharmonic parabolic equations
Decay and local eventual positivity for biharmonic parabolic equations
复制标题
双调和抛物线方程的衰变和局部最终正性
DOI：
10.3934/dcds.2008.21.1129
复制
发表时间：
2008
影响因子：
1.1
通讯作者：
H. Grunau
中科院分区：
文献类型：
--
作者：
A. Ferrero;F. Gazzola;H. Grunau
我们研究了以下问题解的存在性和正性
线性和半线性抛物型方程的柯西问题
以双调和算子为椭圆主元。这个
抛物算子$\PARTIAL_t+\Delta^2$的自相似核
是一个符号变换函数及其演化的解
初始数据为正的问题可能会显示为几乎
符号的瞬间变化。我们根据初始条件确定条件
相应的解表现出某种形式的基准面
积极的行为。我们证明了最终的局部正性
在线性和半线性情况下都是如此。同时，我们展示了
解的负性也可能发生在任意大的情况下
在给定的时间内，只要初始数据被适当地构建。
We study existence and positivity properties for solutions of
Cauchy problems for both linear and semilinear parabolic equations
with the biharmonic operator as elliptic principal part. The
self-similar kernel of the parabolic operator $\partial_t+\Delta^2$
is a sign changing function and the solution of the evolution
problem with a positive initial datum may display almost
instantaneous change of sign. We determine conditions on the initial
datum for which the corresponding solution exhibits some kind of
positivity behaviour. We prove eventual local positivity properties
both in the linear and semilinear case. At the same time, we show
that negativity of the solution may occur also for arbitrarily large
given time, provided the initial datum is suitably constructed.