In this paper, we deal with the biharmonic heat equation gradient non-linearity. Under suitable condition of initial datum, show that global unique existence mild solution. The main technique in paper is to use Banach?s fixed point theorem combination Lp-Lq evaluation operator.

We investigate the Lp-spectrum of linear operators defined consistently on Lp(Ω) for p0 ≤ p ≤ p1, where (Ω, μ) is an arbitrary σ-finite measure space and 1 ≤ p0 < p1 ≤ ∞. We prove p-independence of the Lp-spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω, μ); the balls with respect to this semi-metric are required to satisfy a...

For a general measure space (Ω, μ), it is shown that for every band M in Lp(μ) there exists a decomposition μ = μ ′ + μ′′ such that M = Lp(μ ′) = {f ∈ Lp(μ); f = 0 μ′′-a.e.}. The theory is illustrated by an example, with an application to absorption semigroups. MSC 2010: 46B42, 28A05, 47D06

Enflo and Rosenthal [4] proved that `p(א1), 1 < p < 2, does not (isomorphically) embed into Lp(μ) with μ a finite measure. We prove that if X is a subspace of an Lp space, 1 < p < 2, and `p(א1) does not embed into X, then X embeds into Lp(μ) for some finite measure μ.

A satisfactory representation theory in terms of integrals for linear operators from an Lp space to an arbitrary Banach space X has not as yet been developed for p>l. For the case p = l, Dunford and Pettis [3] have obtained results provided X satisfies certain rather general restrictions. In particular, the representations for the cases in which X is an L space or an Lp space are well known. Th...

We construct classes of von Neumann algebra modules by considering “column sums” of noncommutative Lp spaces. Our abstract characterization is based on an Lp/2-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L2 case, the concept of Lp bimodule (p 6= 2) turns out to be nearly trivial.

The following tree pattern matching problem is considered: Given two unordered labeled trees P and T , find all occurrences of P in T . Here P and T are called a pattern tree and a target tree, respectively. We first introduce a new problem called the pseudo-tree pattern matching problem. Then we show two efficient bit-parallel algorithms for the pseudo-tree pattern matching problem. One runs i...

If 0 < p < 1 we classify completely the linear operators T: Lp -X where X is a p-convex symmetric quasi-Banach function space. We also show that if T: LLo is a nonzero linear operator, then forp < q < 2 there is a subspace Z of Lp, isomorphic to Lq, such that the restriction of T to Z is an isomorphism. On the other hand, we show that if p < q < o, the Lorentz space L(p, q) is a quotient of Lp ...