TY - JOUR
T1 - Uniqueness of piecewise smooth weak solutions of multidimensional degenerate parabolic equations
AU - Tassa, Tamir
N1 - Funding Information:
*Research supported by ONR Grant N00014-92-J-1890.
PY - 1997/6/15
Y1 - 1997/6/15
N2 - We study the degenerate parabolic equation ut + ∇ · f = ∇ · (Q∇u) + g, where (x, t) ∈ ℝN × ℝ+, the flux f, the viscosity coefficient Q, and the source term g depend on (x, t, u) and Q is nonnegative definite. Due to the possible degeneracy, weak solutions are considered. In general, these solutions are not uniquely determined by the initial data and, therefore, additional conditions must be imposed in order to guarantee uniqueness. We consider here the subclass of piecewise smooth weak solutions, i.e., continuous solutions which are C2-smooth everywhere apart from a closed nowhere dense collection of smooth manifolds. We show that the solution operator is L1-stable in this subclass and, consequently, that piecewise smooth weak solutions are uniquely determined by the initial data.
AB - We study the degenerate parabolic equation ut + ∇ · f = ∇ · (Q∇u) + g, where (x, t) ∈ ℝN × ℝ+, the flux f, the viscosity coefficient Q, and the source term g depend on (x, t, u) and Q is nonnegative definite. Due to the possible degeneracy, weak solutions are considered. In general, these solutions are not uniquely determined by the initial data and, therefore, additional conditions must be imposed in order to guarantee uniqueness. We consider here the subclass of piecewise smooth weak solutions, i.e., continuous solutions which are C2-smooth everywhere apart from a closed nowhere dense collection of smooth manifolds. We show that the solution operator is L1-stable in this subclass and, consequently, that piecewise smooth weak solutions are uniquely determined by the initial data.
UR - http://www.scopus.com/inward/record.url?scp=0031153491&partnerID=8YFLogxK
U2 - 10.1006/jmaa.1997.5417
DO - 10.1006/jmaa.1997.5417
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AN - SCOPUS:0031153491
SN - 0022-247X
VL - 210
SP - 598
EP - 608
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -