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In this paper, we study mathematical properties of an integro-differential equation that arises as a particular limit case in the study of individual cell-based model. We obtain global well-posedness for some classes of interaction potentials an[...]texto impreso
This paper deals with the Cauchy problem u(t)-u(xx)+u(p)=0; -infinitytexto impreso
We consider here the homogeneous Dirichlet problem for the equation u(t)= u?u - ?|?u|(2) with ? ? R, u ? 0, in a noncylindrical domain in space-time given by |x| ? R(t) = (T - t)(p), with p > 0. By means of matched asymptotic expansion techniqu[...]texto impreso
In this paper the authors study the asymptotic behaviour of solutions u?(x,t) of the Cauchy problems as ? goes to zero: ut???u+up=0, x?RN, t> 0; u(x,0)=u0(x), x?RN, 0texto impreso
We consider the Cauchy problem u t -u xx +u p =0,x??,t> 0,u(x,0)=u 0 (x),x??, where 0texto impreso
Consider the Cauchy problem u(t) - u(xx) - F(u) = 0; x is-an-element-of R, t> 0 u(x, 0) = u0(x); x is-an-element-of R where u0 (x) is continuous, nonnegative and bounded, and F(u) = u(p) with p > 1, or F(u) = e(u). Assume that u blows up at x =[...]texto impreso
We consider the following nonlinear system of parabolic equations: (1) ut =?u???(u?v), ?vt =?v+u?av for x?B R, t> 0. Here ?,? and a are positive constants and BR is a ball of radius R> 0 in R2. At the boundary of BR, we impose homogeneous Neuma[...]texto impreso
We consider the equation (E) u(t) = ?u + u(p) where x ? R(N) (N ? 1), t > 0, p > 1. We show that if N ? 11 and p > N - 2 (N - 1)1/2/(N - 4) - 2(N - 1)1/2 then there exist radial and positive solutions of (E) which blow up at x = 0, t = Ttexto impreso
Let u be a solution of the Cauchy problem ut=uxx+up, x?R, t> 0, u(x,0)=u0(x), x?R, where p> 1 and u0 is continuous, nonnegative, and bounded. Suppose that u blows up at t=Ttexto impreso
The object of this paper is the study of blowing-up phenomena for the initial-boundary value problem (Pa): ut=uxx+?eu for (x,t)?(0,1)×(0,+?), u(0,t)=asin?t and u(1,t)=0 for t?[0,+?), u(x,0)=u0(x) for x?(0,1), where u0(x) is a continuous and boun[...]texto impreso
This work is concerned with the system (S) {u(t)=Delta u-chi del(u del upsilon) for x is an element of Omega, t> 0 Gamma upsilon(t)=Delta upsilon=Delta upsilon+(u-1) for x is an element of Omega, t> 0 where Gamma; chi are positive constants and [...]texto impreso
We consider the Cauchy problem (1) ut=uxx+up, x?R, t> 0, p> 1, (2) u(x,0)=u0(x),x?R, where u0(x) is continuous, nonnegative and bounded. Assume that the solution u(x,t) of (1), (2) blows up at x=0, t=T. We describe here the generic asymptotic be[...]texto impreso
In this paper we review the theory of cells (particles) that evolve according to a dynamics determined by friction and that interact between themselves by means of suitable potentials. We derive by means of elementary arguments several macroscop[...]texto impreso
The authors consider blow-up for the equation (1) ut=?u+up (x?RN, t> 0), where p> 1 and N> 1. For N> 11and (2) p> (N?2(N?1)1/2)/(N?4?2(N?1)1/2)=p1(N) there exist some radial positive solutions that blow up at x=0, t=Ttexto impreso
We consider the semilinear heat equation with critical power nonlinearity. Using formal. arguments based on matched asymptotic expansion techniques, we give a detailed description of radially symmetric sign-changing solutions, which blow-up at x[...]
