sensitivity analysis with the modified heaviside function for the optimal layout design of multi component systems two kinds of design variables i e pseudo density variables associated with the framework structure and location design variables associated with connected components are involved in the layout design of multi component systems although sensitivities with respect to the first ones can easily be carried out as in topology optimization the semi analytical method sam is often used for sensitivity analysis with respect to the location design variables due to the geometric perturbation of the finite element mesh the latter can then be regarded as a geometric perturbation model gpm in this paper we propose a material perturbation model mpm using fixed finite element fe mesh for sensitivity analysis with respect to location design variables the material discontinuity across the boundary between each component and the framework structure is smoothed approximately by means of a modified heaviside function when a location design variable of a certain component is perturbed attached finite elements to the component boundary are assumed to undertake only a shift of material properties while the finite element mesh itself remains geometrically unchanged as a result analytical sensitivities with respect to location design variables are achieved as easily as for pseudo density variables the computing efficiency is thus improved because the velocity field for the mesh perturbation in the semi analytical scheme is no longer needed the mpm is illustrated by means of numerical tests especially the design optimization of 3d multi component systems

the behavior of the life span for solutions tout δu a x upinrd abstract let t denote the life span of the positive bounded solution u x t to the equation u t u a u u p in r d and u x 0 x where 0 a x c r d 0 x c b r d p 1 and 0 is a parameter depending on a p and d it is possible that t for 0 sufficiently small or that t 0 in which case lim 0 t it is always true that lim t 0 in this paper we investigate the asymptotic behavior of t as 0 in the case that t 0 and as in all cases the asymptotic order depends heavily on a p and d in the case that 0 whereas in the case that it depends only on whether there exists an x 0 with a x 0 x 0 0 or whether the supports of a and are separated by a positive distance

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

weighted lp estimates and fujita exponent for a nonlocal equation abstract we investigate a nonlocal equation t u r n j x y u y t d y j l 1 u x t a x t u p in r n where a is unbounded and j belongs to a weighted space crucial weighted l p and interpolation estimates for the green operator are established by a new method based on the sharp young s inequality the asymptotic behavior of a regular varying coefficients exponential series and the properties of auxiliary functions 1 x 2 b 2 that j and b 2 x b b 2 blow up behaviors are investigated by employing test functions r r instead of principal eigenfunctions global well posedness in weighted l p spaces for the cauchy problem is proved when a x the fujita exponent is shown to be 1 2 n our approach generalizes and unifies nonlocal diffusion equations and pseudoparabolic equations

critical dissipative estimate for a heat semigroup with a quadratic singular potential and critical exponent for nonlinear heat equations abstract we consider a heat semigroup with an inverse square potential and prove critical dissipative estimates including the endpoint case some application to a nonlinear heat equation with an inverse square potential is also discussed

classification of blow up with nonlinear diffusion and localized reaction abstract we study the behaviour of nonnegative solutions of the reaction diffusion equation u t u m x x a x u p in r 0 t u x 0 u 0 x in r the model contains a porous medium diffusion term with exponent m 1 and a localized reaction a x u p where p 0 and a x 0 is a compactly supported symmetric function we investigate the existence and behaviour of the solutions of this problem in dependence of the exponents m and p we prove that the critical exponent for global existence is p 0 m 1 2 while the fujita exponent is p c m 1 if 0 p p 0 every solution is global in time if p 0 p p c all solutions blow up and if p p c both global in time solutions and blowing up solutions exist in the case of blow up we find the blow up rates the blow up sets and the blow up profiles we also show that reaction happens as in the case of reaction extended to the whole line if p m while it concentrates to a point in the form of a nonlinear flux if p m if p m the asymptotic behaviour is given by a self similar solution of the original problem

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

quenching for a parabolic equation with variable coefficient modeling mems technology abstract this paper deals with a parabolic equation with variable coefficient modeling mems technology and we will prove the solution of the problem quenches in finite time in the case without stationary solutions numerical simulations are presented to illustrate the theoretical results

blow up analyses in parabolic equations with anisotropic nonstandard damping source abstract in this paper we consider the nonlinear parabolic problems with anisotropic nonstandard growth conditions and damping terms after obtaining well posedness of solutions we give blow up criteria of solutions through constructing different control functions and generalizing eigenfunction method respectively we also classify global solutions in all scope of the variable exponents moreover the sharp blow up rates blow up time and blow up set are determined which seem to be rarely studied for parabolic problems with anisotropic nonstandard damping sources it is interesting that the asymptotic estimates of blow up solutions rely not only on maxima and minima of the anisotropic exponents but also on the geometry properties of the spatial domain and the scope of variable coefficients

existence and nonexistence of global solutions forut δu a x upinrd abstract we study nonnegative solutions of the equation u t 2 u a x u p in r d t 0 under the assumption that a x 0 is on the order x m for m 2 or that 0 a x c x 2 extending the classical result of fujita and more recent results of bandle and levine and of levine and meier we find a critical exponent p p m d such that if 1 p p then there exist no solutions that are global in time while if p p then there exist both global and nonglobal solutions

global nonexistence for the cauchy problem of some nonlinear reaction diffusion systems we first consider the parabolic equationut 2um a x uq f t x u p w t x t 0 x rn where 2 is the 2 fractional power of the laplacian which for 0 2 stands for impurities and f t x and w t x are given nonnegative functions and find its critical exponent then we consider the criticality for five systems of strongly coupled parabolic equations two of them with nonlinear convective terms our results answer positively some open problems raised recently by bandle deng levine and zhang

global and blow up solutions of superlinear pseudoparabolic equations with unbounded coefficient abstract we investigate positive solutions of pseudoparabolic equations t u t u u v x u p in r n 0 where p 1 and v is a possibly unbounded or singular potential under some rather weak assumptions on the potential we establish the existence of solutions both locally and globally in time within weighted lebesgue spaces for the cauchy problem blow up behavior is also derived using the test function method as a consequence we show that if v x where 0 4 n 2 if n 3 and 0 if n 1 2 then the critical exponent of the cauchy problem is 1 2 n this generalizes the result in the case 0 by cao et al 2009

existence and nonexistence of global solutions for a semilinear reaction diffusion system abstract this paper is concerned with the blow up and global existence of nonnegative solutions to the following cauchy problem u t u v p t 0 x r n v t v a x u q t 0 x r n u x 0 u 0 x v x 0 v 0 x x r n where the constants p q 0 and a x 0 is on the order x m as x m r the fujita critical exponent is determined when m 0 and some results of global existence of solution under some assumptions when m 0 are also obtained the results extend those in escobedo and herrero 1991 9 and indicate that m affects the fujita critical exponent

blow up profiles for a reaction diffusion equation with critical weighted reaction abstract we classify the blow up self similar profiles for the following reaction diffusion equation with weighted reaction u t u m x x x u m posed for x t r 0 t with m 1 and 0 in strong contrast with the well studied equation without the weight that is 0 on the one hand we show that for 0 sufficiently small there exist multiple self similar profiles with interface at a finite point more precisely given any positive integer k there exists k 0 such that for 0 k there are at least k different blow up profiles with compact support and interface at a positive point on the other hand we also show that for sufficiently large the blow up self similar profiles with interface cease to exist this unexpected balance between existence of multiple solutions and non existence of any when 0 increases is due to the effect of the presence of the weight x whose influence is the main goal of our study we also show that for any 0 there are no blow up profiles supported in the whole space that is with u x t 0 for any x r and t 0 t

a priori bounds and complete blow up of positive solutions of indefinite superlinear parabolic problems abstract we study a priori estimates of positive solutions of the equation t u u u a x u p x t 0 satisfying the homogeneous dirichlet boundary conditions here is a bounded domain in r n r p 1 is subcritical a c changes sign and a p satisfy some additional technical hypotheses assume that the solution u blows up in a finite time t and the set x a x 0 is connected using our a priori bounds we show that u blows up completely in at t t and the blow up time t depends continuously on the initial data

nonexistence results for a semilinear heat equation with bounded potentials abstract we study the liouville type theorem for positive solutions to the semilinear heat equation u t u v u p in r n r we show that the nonexistence of positive solutions holds for any p 1 and for any v bounded from below by a positive constant in the superlinear case p 1 we give some sufficient conditions on the potential and the exponent which guarantee the nonexistence of positive solutions in the whole space

grow up for a quasilinear heat equation with a localized reaction abstract we study the behaviour of global solutions to the quasilinear heat equation with a reaction localized u t u m x x a x u p m p 0 and a x being the characteristic function of an interval we prove that there exists an exponent p 0 max 1 m 1 2 such that all global solutions are bounded if p p 0 while for p p 0 all the solutions are global and unbounded in the last case we prove that if p m the grow up rate is different to the one obtained when a x 1 while if p m the grow up rate coincides with that rate but only inside the support of a outside the interval the rate is smaller

the laguerre transform in two variables and its application in partial differential equations an attempt is made to investigate some properties of the laguerre transform in two variables this technique is very effective for solving certain types of partial differential equations

excluding blowup at zero points of the potential by means of liouville type theorems we consider the diffusive hamilton jacobi equation with homogeneous dirichlet conditions and regular initial data it is known from barles dalio 2004 that the problem admits a unique continuous global viscosity solution which extends the classical solution in case gradient blowup occurs we study the question of the possible loss of boundary conditions after gradient blowup which seems to have remained an open problem till now somewhat surprisingly our results show that the issue strongly depends on the initial data and reveal a rather rich variety of phenomena for any smooth bounded domain we construct initial data such that the loss of boundary conditions occurs everywhere on the boundary as well as initial data for which no loss of boundary conditions occurs in spite of gradient blowup actually we show that the latter possibility is rather exceptional more generally we show that the set of the points where boundary conditions are lost can be prescribed to be arbitrarily close to any given open subset of the boundary

critical exponent for parabolic system with time weighted sources in bounded domain abstract this paper considers the time weighted parabolic system u t u e t v p v t v e t u q in bounded domain with r and p q 0 subject to null dirichlet boundary condition the critical fujita curve is determined as p q c 1 max p q 0 1 where 1 is the first eigenvalue of the laplacian as an extension it is observed for another coupled system u t u m u v p v t v n v u q with p q 1 that there is the fujita critical coefficient max m n 1 namely any nontrivial solution blows up in finite time if and only if max m n 1 the studies of critical curves for coupled systems in the current literature are all heavenly relying upon jensen s inequality and the kaplan method for which one has to deal with complicated discussions on the exponents p q being greater or less than 1 differently in the present framework the heat semigroup is introduced to study critical curves for coupled systems where various superlinear and sublinear cases can be treated uniformly by estimates involved this greatly simplifies the arguments for establishing fujita type theorems

on the fujita exponent for a semilinear heat equation with a potential term abstract we consider the existence and nonexistence of positive global solutions for the cauchy problem t u u v x u u p in r n 0 u x 0 x 0 in r n where p 1 and v behaves like x 2 1 o 1 with 0 as x in this paper we determine the so called fujita exponent p for this cauchy problem furthermore for the critical case p p we prove that the cauchy problem has no global positive solutions

blow up results for a class of first order nonlinear evolution inequalities abstract we investigate the non existence of solutions to a class of evolution inequalities in this case as it happens in a relatively small number of blow up studies nonlinearities depend also on time variable t and spatial derivatives of the unknown the present results which in great part do not require any assumption on the regularity of data are completely new and shown with various applications some of these results referring to the problem u t u a x u p f x in r n t 0 include the non existence results of positive global solutions obtained by fujita and others when a 1 and f 0 bandle levine and levine meier when a x m and f 0 pinsky when either f 0 or f 0 and 0 zhang and bandle levine zhang when a 1 and 1

critical exponents of quasilinear parabolic equations abstract in this paper we study the critical exponents of the cauchy problem in r n of the quasilinear singular parabolic equations u t div u m 1 u t s x u p with non negative initial data here s 0 n 1 n 1 m p 1 and n 1 m 1 m 2 s we prove that p c m 1 m 2 s n 1 is the critical exponent that is if 1 p p c then every non trivial solution blows up in finite time but for p p c a small positive global solution exists

well posedness global existence and large time behavior for hardy henon parabolic equations in this paper we study the nonlinear parabolic equation tu u a x u u tu u a x u u t 0 x rn 0 n 1 a r 0 0 min 2 n and with initial value u 0 u 0 we establish local well posedness in lq rn lq rn and in c0 rn c0 rn in particular the value q n 2 q n 2 plays a critical role for 2 n 2 n we show the existence of global self similar solutions with initial values x x x 2 x x x 2 where l rn l rn is homogeneous of degree 00 and is sufficiently small we then prove that if x x x 2 x x x 2 for x x large then the solution is global and is asymptotic in the l l norm to a self similar solution of the nonlinear equation while if x x x x x x for x x large with 2 n 2 n then the solution is global but is asymptotic in the l l norm to et x x et x x the equation with more general potential tu u v x u u v x x l rn is also studied in particular for initial data x x x 2 x x x 2 x x large we show that the large time behavior is linear if vv is compactly supported near the origin while it is nonlinear if vv is compactly supported near infinity

a new critical phenomenon for semilinear parabolic problems abstract we study the cauchy problem of the inhomogeneous semilinear parabolic equations u up ut w 0 onmn 0 with initial valueu0 0 wheremnis a riemannian manifold with possibly nonnegative ricci curvature there is an exponentp which is critical in the following sense when 1 p the problem has a global positive solution for somew w x 0 and someu0 0