Blow-up for Discretizations of a Nonlinear Parabolic Equation With Nonlinear Memory and Mixt Boundary Condition

  •  Camara Zié    
  •  N’gohisse Konan Firmin    
  •  Yoro Gozo    


In this paper, we study the numerical approximation for the following initial-boundary value problem

v_t=v_{xx}+v^q\int_{0}^{t}v^p(x,s)ds, x\in(0,1), t\in(0,T)

v(0,t)=0, v_x(1,t)=0, t\in(0,T)

v(x,0)=v_0(x)>0}, x\in(0,1)

where q>1, p>0. Under some assumptions, it is  shown that the solution of a semi-discrete form of this problem blows up in the finite time and estimate its semi-discrete blow-up time. We also prove that the semi-discrete blows-up time converges to the real one when the mesh size goes to zero. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis.

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