On certain problems for doubly nonlinear parabolic equations and variable-type equations (Q2713922)

Let \(D\) be a bounded open set in \(\mathbb R^n\) with smooth boundary and let \(Q_T = D\times (0,T)\). The following nonlinear parabolic equation is considered in the cylinder \(Q_T\): NEWLINE\[NEWLINE {\mathcal A} \equiv k(u)u_t - \text{div}\bigl(a(\nabla_x u)\bigr) + f(u) = g(x,t), NEWLINE\]NEWLINE where \(k(\xi)\) and \(f(\xi)\) are given continuous functions from \(\mathbb R\) into \(\mathbb R\) and \(a = (a_1,\dots, a_n)\) is a continuous vector field on \(\mathbb R^n\) satisfying the inequalities NEWLINE\[NEWLINE\sum_{i=1}^na_i(\eta)\eta_i \geq C_1|\eta|^p - C_2,\qquad \sum_{i=1}^n|a_i(\eta)|\leq C_3(1 + |\eta|^{p - 1})NEWLINE\]NEWLINE for all \(\eta\in\mathbb R^n\), where \(C_1 > 0\), \(C_2 \geq 2\), \(C_3 > 0\) are some constants and \(p\geq 2\). The equation is known as a doubly nonlinear parabolic equation.NEWLINENEWLINENEWLINEThe author studies the first and second boundary value problems. In particular, the author exposes sufficient conditions on the coefficients of the equation which guarantee the existence of regular solutions to the problems under consideration. The author also discusses solvability of variational inequalities for the above-mentioned equation, where the set of constraints is a cone of nonnegative functions in a Sobolev space. In this case, the function \(k(\xi)\) can change sign. Then it is shown how to arrive at a local existence theorem for a solution to an initial-boundary value problem on the basis of solvability for an appropriate variational inequality.NEWLINENEWLINENEWLINEFinally, the author proves the existence of a \(t\)-periodic solution for a class of nonlinear parabolic equations with variable time direction.