Knudsen type group for time in ℝ and related Boltzmann type equations

We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time [Formula: see text]. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time [Formula: see text] or for time [Formula: see text] for some [Formula: see text] which is independent of the initial value at time 0. Depending on the collision kernel, [Formula: see text] can be arbitrarily small.

Wave breaking phenomena and global existence for the weakly dissipative generalized Camassa-Holm equation

<p style='text-indent:20px;'>In this paper, we mainly study several problems on the weakly dissipative generalized Camassa-Holm equation. We first establish the local well-posedness of solutions by Kato's semigroup theory. We then derive the necessary and sufficient condition of the blow-up of solutions and a criteria to guarantee occurrence of wave breaking. Moreover, when the solution blows up, we obtain the precise blow-up rate. We finally show that the equation has a unique global solution provided the moment density associated with their initial datum satisfies appropriate sign conditions.</p>

Global existence and uniqueness of the solution to a nonlinear parabolic equation

Consider the equation u’ (t) - u + | u |p u = 0, u(0) = u0(x), (1), where u’ := du/dt , p = const > 0, x E R3, t > 0. Assume that u0 is a smooth and decaying function, ||u0|| = sup |u(x, t)|. x E R3 ,t E R+ It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate ||u(x, t)|| < c, where c > 0 does not depend on x, t.

Estimates of Solutions to Nonlinear Evolution Equations

Consider the equation u’(t) = A (t, u (t)), u(0)= U0 ; u' := du/dt (1). Under some assumptions on the nonlinear operator A(t,u) it is proved that problem (1) has a unique global solution and this solution satisfies the following estimate ||u (t)|| < µ (t) -1 for every t belongs to R+ = [0,infinity). Here µ(t) > 0, µ belongs to C1 (R+), is a suitable function and the norm ||u || is the norm in a Banach space X with the property ||u (t) ||’ <= ||u’ (t) ||.

Qualitative analysis of a parabolic–elliptic attraction–repulsion chemotaxis model with logistic source

In this paper, we focus on the qualitative analysis of a parabolic–elliptic attraction–repulsion chemotaxis model with logistic source. Applying a fixed point argument, [Formula: see text]-estimate technique and Moser’s iteration, we derive that the model admits a unique global solution provided the initial cell mass satisfying [Formula: see text] for [Formula: see text] While for [Formula: see text], there are no restrictions on the initial cell mass and the result still holds.

Convergence of Global Solutions to the Cauchy Problem for the Replicator Equation in Spatial Economics

We study the initial-value problem for the replicator equation of theN-region Core-Periphery model in spatial economics. The main result shows that if workers are sufficiently agglomerated in a region at the initial time, then the initial-value problem has a unique global solution that converges to the equilibrium solution expressed by full agglomeration in that region.

Global solution to the incompressible Oldroyd-B model in hybrid Besov spaces

This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant ? ?(0,1). It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in ?Hs ??Bd/2 2,1 with - d/2 < s < d2-1. In particular, if d ? 3, and taking s=0, then ?H0 ? ?Bd/2 2,1 = B d/2 2,1. Since Bt2,? ? Bd/2 2,1 if t > d/2, this result extends the work by Chen and Miao [Nonlinear Anal.,68(2008), 1928-1939].

GLOBAL SOLUTION FOR THE GENERALIZED ANISOTROPIC NAVIER–STOKES EQUATIONS WITH LARGE DATA

We are concerned with 3D incompressible generalized anisotropic Navier– Stokes equations with hyperdissipative term in horizontal variables. We prove that there exists a unique global solution for it with large initial data in anisotropic Besov space.