This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained. 相似文献
For an integer n3 and any positive number , we establish the existence of smooth functions K on n{0} with |K–1|, such that the equation
has a smooth positive solution which blows up at the origin (i.e., u does not have slow decay near the origin). Furthermore, we show that in some situations K can be extended as a Lipschitz function on n. These provide counter-examples to a conjecture of C.-S. Lin when n>4, and a question of Taliaferro.
Mathematics Subject Classification (2000)Primary 35J60; Secondary 53C21 相似文献
This paper deals with the existence and nonexistence of global positive solution to a semilinear reaction-diffusion system with nonlinear boundary conditions.For the heat diffusion case,the necessary and sufficient conditions on the global existence of all positive solutions are obtained.For the general fast diffusion case,we get some conditions on the global existence and nonexistence of positive solutions.The results of this paper fill the some gaps which were left in this field. 相似文献
The three bilinearities for functions are sharply estimated in function spaces associated to the Schrödinger operator . These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.
In this paper we prove uniqueness of positive solutions to logistic singular problems , , 1$">, 0$"> in , where the main feature is the fact that . More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near . This expansion involves both the distance function and the mean curvature of .
Successful demonstration of selective area doping of planar glass samples for monolithic integration of optically passive and active devices on a single chip is presented. Salt solution of erbium was delivered onto pre-sintered germano-silicate samples via a syringe. The samples were then consolidated to form dense glass layers containing regions doped with rare earth. Erbium tri-chloride solution, 0.1 M, was used during the solution doping phase, with the resulting erbium atomic percentage ranging from more than 0.1-0.4%, increasing linearly with the number of drips applied. 相似文献
This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given. 相似文献
High-quality TiO2 thin films have been deposited from aqueous titanium-peroxo solutions via spin coating. The effects of precursor solution pH on the crystallization behavior, morphology, density, and refractive index of the films are reported. From X-ray diffraction measurements, the amorphous as-deposited films are found to crystallize in the anatase phase at 250 °C. Surface and cross-section SEM images reveal that films deposited from an acidic precursor are more uniform and denser than those deposited from a basic precursor. X-ray reflectivity measurements show that films with smooth surfaces and high densities (up to 87% of single-crystal anatase) can be produced at temperatures as low as 300 °C. Measured densities are consistent with high refractive indices at 633 nm of 2.24 and 2.11 for films derived from acidic and basic precursors, respectively. The uniformity and dense nature of the films have allowed fabrication of multilayer dielectric optical elements with thermal processing at only 300 °C. The distributed Bragg reflector with four bilayers exhibits a reflectance of 92% and a stop band width of 150 nm. The optical microcavity has a quality factor of 20. The optical properties of all elements agree well with theoretical models, indicating good optical quality. Use of the precursor chemistry for direct photopatterning of TiO2 films without a polymer resist is also demonstrated. 相似文献