initial trace of solutions of some quasilinear parabolic equations with absorption we study the existence of an initial trace of nonnegative solutions of the problem tu u p 2 u uq 0in qt 0 t we prove that the initial trace is an outer regular borel measure which may not be locally bounded for some values of the parameters p and q we study also the corresponding cauchy problems with a given generalized borel measure as initial data

modularity of simple termination of term rewriting systems with shared constructors abstract kurihara m and a ohuchi modularity of simple termination of term rewriting systems with shared constructors theoretical computer science 103 1992 273 282 a term rewriting system is simply terminating if there exists a simplification ordering showing its termination let r and r be term rewriting systems which share no defined symbol but may share constructors constructors are function symbols which do not occur at the leftmost position in left hand sides of rewrite rules the rest of the function symbols are defined symbols in this paper we prove that r u r is simply terminating if and only if both r and r are so 1 introduction a property p of term rewriting systems trss 4 6 is modular if r u r has the property p if and only if both r0 and r have that property starting with toyama 141 several authors studied modular aspects of trss under the disjointness condition which requires that

modularity in term rewriting revisited we revisit modularity in term rewriting which for the last 25 years has been a very active and fruitful research field starting with the pioneering works of yoshihito toyama on the modularity of confluence and the non modularity of termination he thus initiated an extremely productive line of research with many non trivial and deep results striking counterexamples and a substantial amount of systematic theoretical foundations methodological principles and novel proof techniques in this focused summary we will revisit the modularity analysis for ordinary term rewriting systems considering various confluence and termination properties and restricting ourselves mainly to the case of disjoint unions we will summarize known results on the non modularity of various confluence and termination properties and exhibit crucial ideas constructions and phenomena later on we will also briefly consider various extensions applications related questions and open problems as well as recent developments

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

on termination and confluence properties of disjoint and constructor sharing conditional rewrite systems abstract we investigate the modularity behavior of termination and confluence properties of join conditional term rewriting systems we give counterexamples showing that the properties weak termination weak innermost termination and strong innermost termination are not preserved in general under signature extensions then we develop sufficient conditions for the preservation of these properties under signature extensions and more generally for their modularity in the disjoint union case this leads to new criteria for modularity of termination and completeness generalizing known results for unconditional systems finally combining our analysis with recent related results on the preservation of semi completeness we show how to cover the non disjoint case of combined conditional rewrite systems with shared constructors too

remarks on a nonlinear parabolic equation the equation ut u u r is studied in rn and in the periodic case it is shown that the equation is well posed in l1 and possesses regularizing properties for nonnegative initial data and 0 the solution decays in l1 rn as t in the periodic case it tends uniformly to a limit a consistent difference scheme is presented and proved to be stable and convergent

optimal bounds and blow up phenomena for parabolic problems in narrowing domains we consider degenerate parabolic problems in domains with noncompact boundary and infinite volume in any spatial dimension the equation is of doubly nonlinear type on the boundary we prescribe a homogeneous neumann condition the spatial domain is narrowing at infinity we prove uniform convergence to 0 of solutions as time approaches to this end due to the geometry of the domain the requirement that the initial datum have finite mass is not enough and we have to stipulate the further assumption that a certain moment of the initial datum connected with the geometry of the domain is finite we prove optimal asymptotic estimates of the solution moreover we apply our method to the investigation of blow up problems in narrowing domains obtaining a sharp condition in integral form for the existence of solutions defined for all positive times

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

on the cauchy problem for the evolution p laplacian equations with gradient term and source abstract in this paper the authors study the equation u t div d u p 2 d u u q 1 u d u l in r n with p 2 we first prove that for 1 l p 1 the solution exists at least for a short time then for p 2 l p 1 the existence and nonexistence of global in time solutions are studied in various situations

simple termination of rewrite systems abstract in this paper we investigate the concept of simple termination a term rewriting system is called simply terminating if its termination can be proved by means of a simplification order the basic ingredient of a simplification order is the subterm property but in the literature two different definitions are given one based on strict partial orders and another one based on preorders or quasi orders we argue that there is no reason to choose the second one as the first one has certain advantages simplification orders are known to be well founded orders on terms over a finite signature this important result no longer holds if we consider infinite signatures nevertheless well known simplification orders like the recursive path order are also well founded on terms over infinite signatures provided the underlying precedence is well founded we propose a new definition of simplification order which coincides with the old one based on partial orders in case of finite signatures but which is also well founded over infinite signatures and covers orders like the recursive path order we investigate the properties of the ensuing class of simply terminating systems

on the modularity of termination of term rewriting systems it is well known that termination is not a modular property of term rewriting systems i e it is not preserved under disjoint union the objective of this paper is to provide a uniform framework for sufficient conditions which ensure the modularity of termination we will prove the following result whenever the disjoint union of two terminating term rewriting systems is nonterminating then one of the systems is not ce terminating i e it loses its termination property when extended with the rules cons x y x and cons x y y and the other is collapsing this result has already been achieved by gramlich 7 for finitely branching term rewriting systems a more sophisticated approach is necessary however to prove it in full generality most of the known sufficient criteria for the preservation of termination 24 15 13 7 follow as corollaries from our result and new criteria are derived this paper particularly settles the open question whether simple termination is modular in general we will moreover shed some light on modular properties of combined systems with shared constructors for instance it will be shown that the above result does not carry over to constructor sharing systems

on general best proximity pairs and equilibrium pairs in free abstract economies abstract in this paper using lassonde s fixed point theorem for kakutani factorizable multifunctions and park s fixed point theorem for acyclic factorizable multifunctions we will prove new existence theorems for general best proximity pairs and equilibrium pairs for free abstract economies which generalize the previous best proximity theorems and equilibrium existence theorems due to srinivasan and veeramani p s srinivasan p veeramani on best approximation pair theorems and fixed point theorems abstr appl anal 2003 1 2003 33 47 p s srinivasan p veeramani on existence of equilibrium pair for constrained generalized games fixed point theory appl 2004 1 2004 21 29 and kim and lee w k kim k h lee existence of best proximity pairs and equilibrium pairs j math anal appl 316 2006 433 446 in several aspects

best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces abstract in hyperconvex metric spaces we consider best approximation invariant approximation and best proximity pair problems for multivalued mappings that are condensing or nonexpansive

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

time dependent singularities in semilinear parabolic equations existence of solutions abstract we study the existence of solutions of the semilinear parabolic equation t u u u p in d r n i here n 2 p 1 n 2 p n and i r is an open interval it is assumed that d r n i is supported on t t r n 1 t i for some c 1 2 i r n and is given by i t t d t where is a radon measure on i satisfying a b b a a b i for some positive function in the case where i is bounded we give conditions on the behavior of near 0 for the existence and nonexistence of solutions the existence for i r is also shown under additional conditions on p and the behavior of near moreover the singularity of solutions at x t is examined

existence of best proximity pairs and equilibrium pairs in this paper using the fixed point theorem for kakutani factorizable multifunctions we shall prove new existence theorems of best proximity pairs and equilibrium pairs for free abstract economies which include the previous fixed point theorems and equilibrium existence theorems

modularity in noncopying term rewriting it is known that termination and completeness are not modular properties of term rewriting systems the disjoint union of terminating complete term rewriting systems need not be terminating complete in this paper we introduce a class of noncopying term rewriting systems as a new term based formalism for a kind of graph rewriting systems and prove that this class enjoys the modularity of termination and of completeness actually our results are stronger in the sense that the disjointness condition is relaxed for allowing the systems to share constructors and some defined symbols while preserving the main results

l estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms abstract we study the nonnegative solutions of the viscous hamilton jacobi problem u t u u q 0 u 0 u 0 in q t 0 t where q 1 0 t 0 and r n or is a smooth bounded domain and u 0 l r r 1 or u 0 m b we show l decay estimates valid for any weak solution without any conditions as x and without uniqueness assumptions as a consequence we obtain new uniqueness results when u 0 m b and q n 2 n 1 or u 0 l r and q n 2 r n r we also extend some decay properties to quasilinear equations of the model type u t p u u 1 u u q 0 where p 1 0 and u is a signed solution

on the church rosser property for the direct sum of term rewriting systems the direct sum of two term rewriting systems is the union of systems having disjoint sets of function symbols it is shown that if two term rewriting systems both have the chruch rosser property then the direct sum of these systems also has this property

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

from diagrammatic confluence to modularity this paper builds on a fundamental notion of rewriting theory that characterizes confluence of a binary rewriting relation klop s cofinal derivations cofinal derivations were used by van oostrom to obtain another characterization of confluence of a rewriting relation via the existence of decreasing diagrams for all local peaks in this paper we show that cofinal derivations can be used to give a new concise proof of toyama s celebrated modularity theorem and its recent extensions to rewriting modulo in the case of strongly coherent systems an assumption discussed in depth here this is done by generalizing cofinal derivations to cofinal streams allowing us in turn to generalize van oostrom s result to the modulo case

on the cauchy problem for the singular parabolic equations with gradient term abstract the cauchy problem for a singular parabolic equation with gradient term of the form u t div d u p 2 d u d u q is studied this paper both extends the existence of solutions of the porous media equation with the same gradient term to this cauchy problem for the case q p 1 and obtains the existence of solutions on the assumptions that measures as initial data for the case 0 q p 1 however as q p 1 the optimality of the stipulated assumptions to existence of solutions is still open

cauchy problem for doubly nonlinear parabolic equations with initial data measures abstract we study the existence of solutions for the doubly nonlinear parabolic equation with strongly nonlinear source under optimal assumptions on the initial data which are assumed to be measures we also obtain some local regularity estimates for the supercritical case q m p 1 p n

cauchy problem for fast diffusion equations with a gradient term we study the cauchy problem for fast diffusion equation with a gradient term of the form ut um duq where 0 0 and 0 0 however as q m the optimality of the stipulated assumptions is left open

bubbles in modularity we provide a global technique called neatening for the study of modularity of left linear term rewriting systems objects called bubbles are identified as the responsibles of most of the problems occurring in modularity and the concept of well behaved from the modularity point of view reduction called neat reduction is introduced neatening consists of two steps the first is proving a property is modular when only neat reductions are considered the second is to neaten a generic reduction so to obtain a neat one thus showing that restricting to neat reductions is not limitative this general technique is used to provide a unique uniform method able to elegantly prove all the existing results on the modularity of every basic property of left linear term rewriting systems and also to provide new results on the modularity of termination

existence of solutions to the slow diffusion equation with a nonlinear source abstract in this paper we consider the existence of solutions to cauchy problem for the slow diffusion equation with a source term t u u m u p x r n t 0 u x 0 x 0 where m 1 and p m our purpose is to obtain lower estimates of the existence time of solutions in the uniformly local lebesgue spaces furthermore we obtain the estimate of blow up time blow up rate of solutions and blow up of critical norm

degenerate parabolic equations with initial data measures we address the problem of existence of solutions to degenerate and nondegenerate parabolic equations under optimal assumptions on the initial data which are assumed to be measures the requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation and with the form of the nonlinear forcing term the latter depends on the space gradient of a power of the solution applications to related problems are also outlined

existence of solutions for a fractional semilinear parabolic equation with singular initial data abstract in this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the cauchy problem t u 2 u u p x r n t 0 u 0 0 in r n where n 1 0 2 p 1 and is a radon measure or a measurable function in r n our conditions lead optimal estimates of the life span of the solution with behaving like x a a 0 at the space infinity as 0

cauchy problem for nonlinear parabolic equations with a gradient term we consider the cauchy problem of the nonlinear diffusive hamilton jacobi equation ut um duq where m 0 the existence and nonexistence of local and global solutions are studied by a priori estimates and compactness methods we prove that the cauchy problem of this equation has no nontrivial nonnegative global solutions if 0 moreover we obtain the global existence of solutions in the limiting case and the exact l estimates for local and global solutions

the cauchy problem for ut δu u q with q a positive real number the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces in that context it is known as the generalized deterministic kpz equation the paper is concerned with the initial value problem for the equation under the assumption that the initial data function is bounded and continuous results on the existence uniqueness and regularity of solutions are obtained

on best proximity pair theorems for relatively u continuous mappings abstract a new class of mappings called relatively u continuous is introduced and used to investigate the existence of best proximity points as an application of the existence theorem we obtain a generalized version of the markov kakutani theorem for best proximity points in the setting of a strictly convex banach space

local and global existence for quasilinear parabolic systems with a strongly nonlinear source abstract we study the cauchy problem for quasilinear parabolic systems with a strongly nonlinear source on the assumptions that the initial values are taken in the sense of l l o c 1 r n we obtain the local existence global existence and nonexistence of solutions particularly in the case of global solvability some exact estimates of a solution are obtained

cauchy problem for doubly degenerate parabolic equation with gradient source abstract we consider the cauchy problem for a doubly degenerate parabolic equation with a source term depending solely on the gradient a major result of this paper is to establish the local existence of solutions for this equation under optimal stipulated assumptions on the initial data

singular parabolic equations with measures as initial data abstract in this paper we study the cauchy problem for the singular evolution p laplacian equations with gradient term and source on the assumption of measures as initial conditions for the supercritical case q p 1 p n we obtain that for every nonnegative solution there exists a nonnegative radon measure as initial trace and has some local regularity

convergence and existence results for best proximity points we provide a positive answer to a question raised by eldred and veeramani a a eldred p veeramani existence and convergence of best proximity points j math anal appl 323 2006 1001 1006 about the existence of a best proximity point for a cyclic contraction map in a reflexive banach space moreover we introduce a new class of maps called cyclic contractions which contains the cyclic contraction maps as a subclass convergence and existence results of best proximity points for cyclic contraction maps are also obtained 2008 elsevier ltd all rights reserved

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

quasi linear diffusion equations with gradient terms and l1 data abstract in this article we study the following quasi linear parabolic problem u t u u 2 u u q u 2 u u p in 0 t u x t 0 on 0 t u x 0 u 0 x in where is a bounded open set of r n and t 0 we prove that if 1 0 p u 0 l 1

existence and convergence of best proximity points abstract consider a self map t defined on the union of two subsets a and b of a metric space and satisfying t a b and t b a we give some contraction type existence results for a best proximity point that is a point x such that d x t x dist a b we also give an algorithm to find a best proximity point for the map t in the setting of a uniformly convex banach space

on the cauchy problem for the evolution p laplacian equations with gradient term and source and measures as initial data abstract in this paper we study the cauchy problem for the evolution p laplacian equations with gradient term and source on the assumption that measures as initial conditions