Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 2.9 ) Pub Date : 2021-03-17 , DOI: 10.1007/s13398-021-01010-w Martina Magliocca
We want to analyze both regularizing effect and long, short time decay concerning a class of parabolic equations having first order superlinear terms. The model problem is the following:
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle u_t-\text {div }(A(t,x)|\nabla u|^{p-2}\nabla u)=\gamma |\nabla u|^q &{} \text {in}\,\,(0,T)\times \Omega ,\\ u=0 &{}\text {on}\,\,(0,T)\times \partial \Omega ,\\ u(0,x)=u_0(x) &{}\text {in}\,\, \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \) is an open bounded subset of \({{\,\mathrm{{{\mathbb {R}}}}\,}}^N\), \(N\ge 2\), \(0<T\le \infty \), \(1<p<N\) and \(q<p\). We assume that A(t, x) is a coercive, bounded and measurable matrix, the growth rate q of the gradient term is superlinear but still subnatural, \(\gamma \) is a positive constant, and the initial datum \(u_0\) is an unbounded function belonging to a well precise Lebesgue space \(L^\sigma (\Omega )\) for \(\sigma =\sigma (q,p,N)\).
中文翻译:
具有排斥性超线性一阶项的抛物线问题的正则化效应和衰减结果
我们想分析关于一阶具有一阶超线性项的抛物方程的正则化效应和长短时间衰减。模型问题如下:
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ displaystyle u_t- \ text {div}(A(t,x)| \ nabla u | ^ {p-2} \ nabla u )= \ gamma | \ nabla u | ^ q&{} \ text {in} \,\,(0,T)\ times \ Omega,\\ u = 0&{} \ text {on} \,\, (0,T)\ times \ partial \ Omega,\\ u(0,x)= u_0(x)&{} \ text {in} \,\,\ Omega,\ end {array} \ right。} \ end {aligned} $$其中\(\ Omega \)是\({{\,\ mathrm {{{\ mathbb {R}}}} \,}} ^ N \),\(N \ ge 2 \),\(0 <T \ le \ infty \),\(1 <p <N \)和\(q <p \)。我们假设A(t, x)是一个强制性,有界且可测量的矩阵,梯度项的增长率q是超线性的,但仍然是亚自然的,\(\ gamma \)是一个正常数,初始数据\(u_0 \)是属于良好精确勒贝格空间无界函数\(L ^ \西格马(\欧米茄)\)为\(\西格玛= \西格马(q,p,N)\) 。