Asymptotic behavior result for obstacle parabolic problems with measure data
- PDF / 979,041 Bytes
- 37 Pages / 439.37 x 666.142 pts Page_size
- 46 Downloads / 203 Views
Advances in Operator Theory https://doi.org/10.1007/s43036-020-00113-2
(0123456789().,-volV)(0123456789().,-volV)
ORIGINAL PAPER
Asymptotic behavior result for obstacle parabolic problems with measure data M. Abdellaoui1 Received: 4 August 2020 / Accepted: 30 September 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract The paper deals with the asymptotic behavior, as n tends to þ1, of distributional solutions for a class of variational parabolic inequalities of the following form: 8Z T > > > hðun Þt ; un wi þ hdiv aðt; x; run Þ; un wiW 1;p0 ðXÞ;W 1;p ðXÞ dt > > 0 > 0 > > > > > < hl; un wiMb ðQÞ;C0 ðQÞ ; 8un ; w 2 Kn ; n ðPÞ 1;p > p p0 1;p0 > ðXÞÞ; zð0; xÞ ¼ 0; > with Kn ¼ z 2 L ð0; T; W0 ðXÞÞ such that zt 2 L ð0; T; W > > > > o > > > > 1;1 :¼ krzk ¼ sup krzk n and kzk 1 1 1 L ðQÞ L ðXÞ : L ð0;T;W0 ðXÞÞ t2½0;T
where X is a bounded open set in RN (N 2), T [ 0, l is a nonhomogeneous Radon measure and u7! div aðt; x; run Þ is a defined operator satisfying Leray–Lions assumptions. We provide a proof of the convergence of un (solution of this problem) to the distributional solution u of the corresponding homogeneous parabolic equation ut div aðt; x; ruÞ ¼ l in Q with uð0; xÞ ¼ 0 and the same datum l. Keywords Asymptotic behavior Capacity and measures Cut-off functions Truncations Variational inequalities
Mathematics Subject Classification 28A12 35B40 35B65 35R45
Communicated by Mark Veraar. & M. Abdellaoui [email protected] 1
LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, B.P. 1796, Atlas Fez, Morocco
M. Abdellaoui
1 Introduction Let Q ¼ ð0; TÞ X be a parabolic domain where X is a bounded open subset of RN ðN 2Þ and T is a positive number; we are mainly concerned with the asymptotic behaviour of solutions for a class of ‘‘variational parabolic inequalities’’ modeled by 8Z T > > > hðun Þt ; un wi þ hdiv aðt; x; run Þ; un wiW 1;p0 ðXÞ;W 1;p ðXÞ dt > > 0 > 0 > > > > > < hl; un wiMb ðQÞ;C0 ðQÞ ; 8un ; w 2 Kn ; n 0 0 > > with K ¼ z 2 Lp ð0; T; W01;p ðXÞÞ s.t. zt 2 Lp ð0; T; W 1;p ðXÞÞ; zð0; xÞ ¼ 0; > n > > > > o > > > > 1;1 :¼ krzk ¼ sup krzk n ; kzk 1 1 1 L ðQÞ L ðXÞ : L ð0;T;W0 ðXÞÞ t2½0;T
ð1Þ 1 where 2 Nþ1 p\N, the function a : ð0; TÞ X RN 7!RN is a vector field which satisfies the classical ‘‘Leray–Lions’’ assumptions (the divergential operator u7! div aðt; x; run Þ defines a strictly monotone operator from Lp ð0; T; W01;p ðXÞÞ 0 0 to Lp ð0; T; W 1;p ðXÞÞ) and l 2 Mb ðQÞ is a general Radon measure with bounded N total variation jljðQÞ. The lower bound on p is required since p Nþ1 [ 1 if and 1 only if p [ 2 Nþ1 (for smaller values of p we cannot even use the framework of Sobolev spaces to deal with (1) and then it cannot be solved in the variational setting). First, in order to deal with problems with regular/irregular data, ‘‘distributional’’ solutions are defined in [49] in order to get a variational theory for elliptic/parabolic equations. Afterwards, ‘‘entrop
Data Loading...
