On the Dirichlet problem for fully nonlinear parabolic equations
Let \(Q=\Omega\times (0,T)\) with \(\Omega\) a bounded domain of \(\mathbb{R}^n\). The paper concerns the solvability of the first initial-boundary value problem: \(G[u]=g\) in \(Q\), \(u(x,0)=\psi(x)\), and \(u(x,t)=\Phi\) on \(\partial\Omega \times [0,T]\), where \(G\) is a second order fully nonlinear parabolic operator defined as follows. Let \(G(s,S)\) be a function defined on \(D_0\subset \mathbb{R}^1 \times S^n\), where \(S^n\) is the set of \(n\times n\) symmetric matrices. Assume there exists a nonempty convex domain \(D\) where \(G(s,S)\) is convex and monotone in a suitable way. Indeed, two types of functions are considered. One type is: \(G(s,S)=-s+F(s)\), \(D=\mathbb{R}^1\times K(F)\), where \(K(F)\) is a component of the concavity and monotonicity of \(F\). One typical example is: \(F(s)=(\text{tr}_mS/\text{tr}_lS)^{1/(m-l)}\), \(0\leq l < m\leq\), \(K(F)=\{s\in S^n: \text{tr}_p S>0\), \(p=1,\ldots, m\}\). The second type discussed is: \(G(s,S)=(s \text{tr}_m S)^{-1}\), \(D=(-\infty,0)\times F(K)\). The function \(G(s,S)\) determines fully nonlinear parabolic operators in the following way: \(G[u]=G(a_0u_t+w,A(u_x)u_{xx}A(u_x)-a_1I +W)\), where \(a_0, a_1\geq 0\), \(w\) is a smooth function, \(W\) is a matrix, \(A(u_x)\) is a smooth positive matrix, and \(u_{xx}\) is the Hessian matrix of \(u\). The peculiarity of the equation considered is the nonlinear dependence both on the first order time derivative and the second order spatial ones. The solvability of the problems depend on the coefficients \(a_0\) and \(a_1\). They are classified as: evolution of open type if \(a_0>0, \;a_1=0\); evolution of closed type if \(a_0=0, \;a_1>0\); evolution of mixed type if \(a_0>0, \;a_1>0\). In order to have admissible solutions, the compatibility of \(\psi, \Phi\) are discussed. In contrast to the linear case, the initial velocity \(v(x)=u_t(x,0)\) must satisfy a nonlinear equation. The solvability of the general problem is investigated by using a suitable continuity method. Some a priori estimates necessary to use the continuity method are discussed. Also the uniqueness problem is studied.
second-order fully nonlinear parabolic operator
scientific article; zbMATH DE number 1348498
scientific article