The Fujita exponent for the Cauchy problem in the hyperbolic space

It is well known that the heat kernel in the hyperbolic space has a different behavior for large times than the one in the Euclidean space. The main purpose of this paper is to study its effect on the positive solutions of Cauchy problems with power nonlinearities. Existence and non-existence results for local solutions are derived. Emphasis is put on their long time behavior and on Fujita?s phenomenon. To have the same situation as for the Cauchy problem in RN, namely finite time blow up for all solutions if the exponent is smaller than a critical value and existence of global solutions only for powers above the critical exponent, we must introduce a weight depending exponentially on the time. In this respect the situation is similar to problems in bounded domains with Dirichlet boundary conditions. Important tools are estimates for the heat kernel in the hyperbolic space and comparison principles.     

Multiple blow-up rates in a coupled heat system with mixed-type nonlinearities

This paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed-type nonlinearities. At first, the critical exponent is obtained, and simply described via a characteristic algebraic system introduced by us. Then, as the main results of the paper, three blow-up rates are established under different dominations of nonlinearities for the one-dimensional case, and represented in another characteristic algebraic system. In particular, it is observed that unlike those in previous literature on parabolic models with absorptions, two of the multiple blow-up rates obtained here do depend on the absorption exponents. In the known works, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates. To our knowledge, this is the first example of absorption-dependent blow-up rates, exploiting the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system. It is also proved that the blow-up of solutions in the model occurs on the boundary only.     

Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources

This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depends on the spacial dimension. Exactly speaking, the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this threshold. Supported by the NNSF of China and the China Postdoctoral Science Foundation.     

On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent

This paper is concerned with a semilinear parabolic equation involving critical Sobolev exponent in a ball or in RN. The asymptotic behavior of unbounded, radially symmetric, nonnegative global solutions which do not decay to zero is given. The structure of the space of initial data is also discussed.     

Support properties of solutions to a degenerate equation with absorption and variable density

In this paper, we study the properties of solutions to a degenerate parabolic equation with variable density and absorption. We first obtain a critical exponent, which distinguishes the localization of solutions from the positivity of them. When positivity prevails, we obtain the other critical exponent with respect to the decay of the variable density, which separates the global existence of interfaces from the disappearance of them. Moreover, the long time behavior of interfaces is characterized.     

Contraction of convex hypersurfaces in Euclidean space

We consider a class of fully nonlinear parabolic evolution equations for hypersurfaces in Euclidean space. A new geometrical lemma is used to prove that any strictly convex compact initial hypersurface contracts to a point in finite time, becoming spherical in shape as the limit is approached. In the particular case of the mean curvature flow this provides a simple new proof of a theorem of Huisken.This work was carried out while the author was supported by an Australian Postgraduate Research Award and an ANUTECH scholarship.     

Cauchy problems of semilinear pseudo-parabolic equations

This paper concerns with the Cauchy problems of semilinear pseudo-parabolic equations. After establishing the necessary existence, uniqueness and comparison principle for mild solutions, which are also classical ones provided that the initial data are appropriately smooth, we investigate large time behavior of solutions. It is shown that there still exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations.     

Continuous spin models on annealed generalized random graphs

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution.First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case.We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in Dommers et al. (2016). On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.     

Asymptotic behavior of solutions to Euler–Poisson equations for bipolar hydrodynamic model of semiconductors

In this paper we study the Cauchy problem for 1-D Euler–Poisson system, which represents a physically relevant hydrodynamic model but also a challenging case for a bipolar semiconductor device by considering two different pressure functions and a non-flat doping profile. Different from the previous studies (Gasser et al., 2003 [7], Huang et al., 2011 [12], Huang et al., 2012 [13]) for the case with two identical pressure functions and zero doping profile, we realize that the asymptotic profiles of this more physical model are their corresponding stationary waves (steady-state solutions) rather than the diffusion waves. Furthermore, we prove that, when the flow is fully subsonic, by means of a technical energy method with some new development, the smooth solutions of the system are unique, exist globally and time-algebraically converge to the corresponding stationary solutions. The optimal algebraic convergence rates are obtained.     

On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudosolutions

Here we examine a method of selective smoothing in image restoration in BV space using a variable exponent functional of linear growth. Variable exponent growth was suggested in P. Blomgren et al. (1997) [6], and in Y. Chen et al. (2006) [8] the notion of a pseudosolution of the minimization problem is discussed, as well as the proof of existence and uniqueness of the solution of the time flow for the associated time dependent PDE. Here we prove stability for pseudosolutions, while also discussing the case for more general convex functionals of linear growth and variable exponent for use as an approximation for the time flow other than that which is used in Y. Chen et al. (2006) [8].     

An equation whose Fujita critical exponent is not given by scaling

In this paper, we prove sharp blow up and global existence results for a heat equation with nonlinear memory. It turns out that the Fujita critical exponent is not the one which would be predicted from the scaling properties of the equation.     

Effects of reactive gradient term in a multi-nonlinear parabolic problem

This paper deals with parabolic equation ut=Δu+r|∇u|−aepu subject to nonlinear boundary flux ∂u/∂η=equ, where r>1, p,q,a>0. There are two positive sources (the gradient reaction and the boundary flux) and a negative one (the absorption) in the model. It is well known that blow-up or not of solutions depends on which one dominating the model, the positive or negative sources, and furthermore on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions. We at first determine the critical blow-up exponent, and then obtain the blow-up rate, the blow-up set as well as the spatial blow-up profile for blow-up solutions in the one-dimensional case. It turns out that the gradient term makes a substantial contribution to the formation of blow-up if and only if r?2, where the critical r=2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level. In addition, it is observed that the gradient term with r>2 significantly affects the blow-up rate also. In fact, the gained blow-up rates themselves contain the exponent r of the gradient term. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. As for the influence of gradient perturbations on spatial blow-up profiles, we only need some coefficients related to r for the profile estimates, while the exponent of the profile itself is r-independent. This seems natural for boundary blow-up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity.     

Boundedness vs. blow-up in a chemotaxis system

We determine the critical blow-up exponent for a Keller-Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.     

Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations

The purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones. In the random case, a notion of principal Lyapunov exponent serving as an analog of principal eigenvalue is introduced. It is shown that the principal Lyapunov exponent is deterministic and of simple multiplicity. It is also shown that there is a one-dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign, which serves as an analog of principal eigenfunction. In addition, monotonicity of the principal Lyapunov exponent with respect to the zero-order terms both in the equation and in the boundary condition is proved. When the second- and first-order terms are deterministic, it is proved that the principal Lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time-averaged equation. In the general nonautonomous case, the concepts of principal spectrum, which serves as an analog of principal eigenvalue, and principal Lyapunov exponents are introduced. As is known, the principal spectrum is a compact interval. It is proved in the paper that the principal spectrum contains all the principal Lyapunov exponents. When the second and first-order terms are time independent, a lower estimate of the infimum of the principal spectrum is given in terms of an associated time-averaged equation.     

Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux

In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation u_t=(log^σ(1+u)u_x)_x, in R₊×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions.     

Blowup in infinite time of radial solutions for a parabolic–elliptic system in high-dimensional Euclidean spaces

We consider radial solutions blowing up in infinite time to a parabolic–elliptic system in N$N$-dimensional Euclidean space. The system was introduced to describe the gravitational interaction of particles. In the case where N≥2$N\ge 2$, we can find positive and radial solutions blowing up in finite time. In the present paper, in the case where N≥11$N\ge 11$, we find positive and radial solutions blowing up in infinite time and investigate those blowup speeds, by using the so-called asymptotic matched expansion techniques and parabolic regularity.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

Large-Time Behaviour of Solutions of the Exterior-Domain Cauchy–Dirichlet Problem for the Porous Media Equation with Homogeneous Boundary Data

The title of this paper states precisely what the subject is. The first part of the paper concerns the radially-symmetric problem in the exterior of the unit ball. It is shown that in time the solution of the problem converges to one of two specific self-similar solutions of the porous media equation, dependent upon the dimensionality of the problem. Moreover, the free boundary of the solution converges to that of the self-similar solution. The critical space dimension is two, for which there is no distinction between the self-similar solutions, and the form of the convergence is exceptional. The technique used is a comparison principle involving a variable that is a weighted integral of the solution. The second part of the paper is devoted to the problem in an arbitrary spatial domain with no conditions of symmetry. A special invariance principle and the results obtained for the radially-symmetric case are used to determine the large-time behaviour of solutions and their free boundaries. This behaviour is decidedly different from when the boundary data are fixed and not homogeneous.     

Blow-up properties of solutions for a multi-coupled parabolic system

This paper deals with the blow-up behavior of radial solutions to a parabolic system multi-coupled via inner sources and boundary flux. We first obtain a necessary and sufficient condition for the existence of non-simultaneous blow-up, and then find five regions of exponent parameters where both non-simultaneous and simultaneous blow-up may happen. In particular, nine simultaneous blow-up rates are established for different regions of parameters. It is interesting to observe that different initial data may lead to different simultaneous blow-up rates even with the same exponent parameters.     

On a coupled nonlinear singular thermoelastic system

For a coupled nonlinear singular system of thermoelasticity with one space dimension, we consider its initial boundary value problem on an interval. For one of the unknowns a classical condition is replaced by a nonlocal constraint of integral type. Because of the presence of a memory term in one of the equations and the presence of a weighted boundary integral condition, the solution requires a delicate set of techniques. We first solve a particular case of the given nonlinear problem by using a functional analysis approach. On the basis of the results obtained and an iteration method we establish the well-posedness of solutions in weighted Sobolev spaces.