6533b85ffe1ef96bd12c0ffd
RESEARCH PRODUCT
Existence and uniqueness for a degenerate parabolic equation with 𝐿¹-data
José M. MazónFuensanta AndreuSergio Segura De LeónJulián Toledosubject
Pure mathematicsMonotone polygonApplied MathematicsGeneral MathematicsOperator (physics)Mathematical analysisDegenerate energy levelsBoundary (topology)Parabolic cylinder functionFunction (mathematics)UniquenessLaplace operatorMathematicsdescription
In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L 1 ( Ω ) L^{1}(\Omega ) , u t = d i v a ( x , D u ) in ( 0 , ∞ ) × Ω , \begin{equation*}u_{t} = \mathrm {div} \mathbf {a} (x,Du) \quad \text {in } (0, \infty ) \times \Omega , \end{equation*} − ∂ u ∂ η a ∈ β ( u ) on ( 0 , ∞ ) × ∂ Ω , \begin{equation*}-{\frac {{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text {on } (0, \infty ) \times \partial \Omega ,\end{equation*} u ( x , 0 ) = u 0 ( x ) in Ω , \begin{equation*}u(x, 0) = u_{0}(x) \quad \text {in }\Omega ,\end{equation*} where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, ∂ / ∂ η a \partial / {\partial \eta _{a}} is the Neumann boundary operator associated to a \mathbf {a} , D u Du the gradient of u u and β \beta is a maximal monotone graph in R × R {\mathbb {R}}\times {\mathbb {R}} with 0 ∈ β ( 0 ) 0 \in \beta (0) .
year | journal | country | edition | language |
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1999-01-01 | Transactions of the American Mathematical Society |