Quasilinear degenerate evolution equations in Banach spaces
The paper is concerned with the Cauchy problem for the quasilinear degenerate evolution equation of parabolic type in a Banach space \(X\): \[ {{dMv}\over{dt}} +L(Mv)v=F(Mv), \quad \;Mv(0)=u_0, \quad 0<t\leq T, \tag{1} \] where \(L(u)\) and \(M\) are closed linear operators with \(D(L(u))=D_L\) for \(u\in K=\{u\in Z: \| u\| _Z\leq R\}\), \(Z\) is a Banach space continuously embedded in \(X\), \(D(M)\supset D_L\), \(M(D_L)\supset Z,\) and \(F: K\to X\) is a nonlinear operator. Existence and uniqueness of an local solution to (1) is under consideration. For this purpose putting \(Mv=u\) the equation is considered in the form of inclusion of nondegenerate type (usually used by the authors) with multivalued linear operator \(A(u)=L(u)M^{-1}\) and nonlinear \(F\): \[ {{du}\over{dt}} +A(u)u\ni F(u), \quad u(0)=u_0, \quad 0<t\leq T. \] On the basis of investigating the inclusion main result obtained is the following. Under the conditions on \(M\)-modified resolvent: \[ \| M(\lambda M -L(u))^{-1}\| _{{\mathcal L}(X)}\leq {{C}\over{(| \lambda| +1)^ {1-k}}}, \quad \lambda\not\in\{\lambda\in {\mathbb C}: | \arg \lambda| <\phi\}, \;u\in K, \;0\leq k<1, \] \[ \| M(\lambda M -L(u))^{-1}\| _{{\mathcal L}(X, Z)}\leq {{C}\over{(| \lambda| +1)^ {1-\rho}}}, \quad \lambda\leq 0, \;u\in K, \;k\leq \rho<1, \] and the Lipschitz condition for \(L(u)\) and \( F\) there exists a unique solution to (1) in the function space that is indicated in the paper.
scientific article; zbMATH DE number 2133421