complete blow up after tmax for the solution of a semilinear heat equation soit un sous ensemble ouvert borne de r n a frontiere lisse on considere le probleme u t u f u dans o t u 0 sur o t u x o u 0 x x ou f r r est localement de lipschitz non decroissante et f o 0

convergence in a quasilinear parabolic equation with time almost periodic boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t g t t 0 where g t is an almost periodic function we prove the following convergence results if a time global solution u is bounded then it converges to an almost periodic solution which is unique up to space shift stable and asymptotically stable if u goes to infinity with positive lower average speed then it converges to an almost periodic traveling wave which is unique up to space shift stable asymptotically stable and has positive average speed

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

convergence in a quasilinear parabolic equation with neumann boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t k t u 1 t t 0 where k are smooth functions which are periodic in both t and u brunovsk et al proved in their paper brunovsk et al 1992 8 that if a time global solution u is bounded then it converges to a periodic solution we prove that if u is unbounded from above then it converges to a periodic traveling wave v x t c t in case k k t or k k u where v is a time periodic function and c 0 in addition the periodic traveling wave is unique up to space shifts or time shifts it is stable and asymptotically stable the average traveling speed c and the instantaneous speed v t c are also studied

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

blow up of solutions of nonlinear heat equations on considere des solutions de u t u xx f u pour a 0 u a t 0 pour t 0 u x 0 x pour a x a ou f u est une fonction superlineaire continument differentiable

on minimal sets of scalar parabolic equations with skew product structures skew product semi flow i t x x y x x y which is generated by ut uxx f y t ix u ux t 0 0 x 1 y e y d or n boundary conditions is considered where x is an appropriate subspace of h2 0 1 y r is a minimal flow with compact phase space it is shown that a minimal set e c x x y of ilt is an almost 1 1 extension of y that is set yo y e y i card e c p 1 y i is a residual subset of y where p x x yy is the natural projection consequently if y r is almost periodic minimal then any minimal set e c x x y of ilt is an almost automorphic minimal set it is also proved that dynamics of it is closed in the category of almost automorphy that is a minimal set e c x x y of fit is almost automorphic minimal if and only if y r is almost automorphic minimal asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed examples of nonalmost periodic almost automorphic minimal sets are provided

renormalization in the henon family i universality but non rigidity in this paper geometric properties of infinitely renormalizable real henon like maps f in mathbb r 2 are studied it is shown that the appropriately defined renormalizations r n f converge exponentially to the one dimensional renormalization fixed point the convergence to one dimensional systems is at a super exponen tial rate controlled by the average jacobian and a universal function a x it is also shown that the attracting cantor set of such a map has hausdorff dimension less than 1 but contrary to the one dimensional intuition it is not rigid does not lie on a smooth curve and generically has unbounded geometry

slow decay of solutions in a semilinear dissipative parabolic equation this paper is concerned with a cauchy problem p u t u x x u p 1 u in r x 0 u x 0 u 0 x in r where p 1 and u0 e l r a solution u of p is said to decay fast as t if limt tl p 1u x t 0 uniformly in r and to decay slowly as t otherwise we prove that if u0 x does not decay faster than x q with some q 0

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

on entropy and monotonicity for real cubic maps consider real cubic maps of the interval onto itself either with positive or with negative leading coefficient this paper completes the proof of the monotonicity conjecture which asserts that each locus of constant topological entropy in parameter space is a connected set the proof makes essential use of the thesis of christopher heckman and is based on the study of bones in the parameter triangle as defined by tresser and r mackay

global blow up of a nonlinear heat equation solutions to semilinear parabolic equations of the form u t u f u x in which blow up at some finite time t are investigated for slowly growing functions f for nonlinearities such as f s 2 s ln 2 s 1 b with 0 u becomes infinite throughout as t t it is alsofound that for marginally more quickly growing functions e g f s 2 s ln 2 s 2 u is unbounded on some subset of which has positive measure and is unbounded throughout if is a small enough region

acoustical cavity excitation a tutorial exposition is given of cavity excitations and it is shown that relatively simple analytical solutions for the excitation of acoustic fields in cavities with lossy walls can be obtained for a broad class of such cavities the standard analytic procedure whereby one expresses the acoustic field as a sum over the eigenmodes of the cavity is reviewed and its limitations are pointed out an alternate procedure is to seek to make use of green s functions depending on a single coordinate and to not express such green s functions as sums over eigenfunctions instead one expresses the green s function as a closed form function that has a discontinuous slope at the value of the coordinate corresponding to the source location although the class of cavities for which this approach is possible is limited it is sufficiently broad to be of intrinsic interest and the results may be helpful to those seeking benchmark solutions and analytic insight displayed examples are for cavities whose shapes are spheres and cylinders with interior monopole dipole line and distributed sources the formulation yields simple expressions for the total power absorbed by the cavity walls

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

nonexistence of wandering domains for strongly dissipative infinitely renormalizable henon maps at the boundary of chaos this article generalizes the nonexistence of wandering domains from unimodal maps to infinitely period doubling renormalizable henon like maps in the strongly dissipative area contracting regime this solves an open problem proposed by van strien discrete contin dyn syst 27 2 557 588 2010 and lyubich and martens invent math 186 1 115 189 2011 we partition the phase space of a henon like map into two regions the good region and the bad region the good region is where the method of proof for unimodal maps applies to henon like maps while the bad region is where serious difficulties occur these difficulties are resolved by the two row lemma an inequality that relates the contraction of areas to the contraction of bad regions after analyzing the competition of the two types of contraction we show that the case of bad regions happens at most finitely many times and complete the proof as an application the theorem enriches our understanding of the topological structure of the heteroclinic web the union of the stable manifolds of periodic orbits forms a dense set in the domain

on a simple maximum principle technique applied to equations on the circle abstract for a class of general quasilinear equations on s 1 we show that by a very simple maximum principle technique as long as the solution stays finite all of its derivatives also remain finite some specific examples are given under suitable assumptions we also derive exponential decay of the derivatives of the solution

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

new explicit solutions in acoustics of closed spaces on the basis of divergent series a new approach to the acoustics of closed spaces is developed that involves solutions for polygonal shapes in explicit form it is shown that exact solutions can be constructed for polygonal geometries where all the interior angles are equal to n n is an integer it is stated that the set of such polygons consists of the rectangle known result and three types of triangles some new explicit formulas are obtained for the eigenfrequencies of the triangles it is demonstrated that the proposed technique also permits an exact representation of the impulse response function for the geometries described

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

analytic solutions of some nonlinear diffusion equations on considere u t u u 2 x n u 0 avec des fonctions initiales non constantes est un domaine borne de r n a frontiere lisse

volume preserving mean curvature flow of revolution hypersurfaces in a rotationally symmetric space in an ambient space with rotational symmetry around an axis which include the hyperbolic and euclidean spaces we study the evolution under the volume preserving mean curvature flow of a revolution hypersurface m generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces tgh for short requiring that for each time t 0 the evolving hypersurface m t meets such tgh orthogonally we prove that a the flow exists while m t does not touch the axis of rotation b throughout the time interval of existence b1 the generating curve of m t remains a graph and b2 the averaged mean curvature is double side bounded by positive constants c the singularity set if non empty is finite and lies on the axis d under a suitable hypothesis relating the enclosed volume to the n volume of m we achieve long time existence and convergence to a revolution hypersurface of constant mean curvature

extension of the image model to arbitrary polyhedra in this paper the image model is extended to arbitrary polyhedra with any number of sides this generalization makes it possible to model real concert halls much more accurately the image positions computed by the method can be reduced to the directional impulse response in order to create audible simulations of the concert hall also the image model can provide insight into the fundamental acoustical properties of different concert hall geometries for example the extended image model demonstrates that rectangular halls have an advantage over fan shaped halls with regard to spatial impression we also show why the image model has fundamental advantages over another popular modeling technique ray tracing

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0