We show that the nonlinear fourth order degenerate parabolic equation u t + div (u n ru) = 0; n > 0 admits nonnegative solutions to initial data which are a nonnegative Radon measure provided that n < 2. In addition, we prove that the equation has a regularizing eeect in the sense that the solution we construct is in H 1 (R N) for all positive times and in H 2 loc (R N) for almost all positive ...

Let Ω ⊂ R be a smooth bounded domain. We give sufficient conditions (which are also necessary in many cases) on two nonnegative functions a, b that are possibly discontinuous and unbounded for the existence of nonnegative solutions for semilinear Dirichlet periodic parabolic problems of the form Lu = λa (x, t) u − b (x, t) u in Ω × R, where 0 < p, q < 1 and λ > 0. In some cases we also show the...

Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an expon...

A new doubling algorithm – Alternating-Directional Doubling Algorithm (ADDA) – is developed for computing the unique minimal nonnegative solution of an M -Matrix Algebraic Riccati Equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms – SDA of Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 393–412) a...

We propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time-varying dimensions. The first method belongs to the family of square-root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations. An alternative balancing-free square-root method has the advantage of a pote...

We study the nonlinear elliptic boundary value problem Au = f(x, u) in Ω , Bu = g(x, u) on ∂Ω , where A is an operator of p−Laplacian type, Ω is an unbounded domain in R with non-compact boundary, and f and g are subcritical nonlinearities. We show existence of a nontrivial nonnegative weak solution when both f and g are superlinear. Also we show existence of at least two nonnegative solutions ...