The numerical approximation of problems with blow-up is studied. In particular, the long-time behaviour of solutions of the semidiscretization in space of the following parabolic problem: \[ \begin{cases} u_t= u_{xx}-\lambda u^p\quad &\text{in }(0,1)\times [0,T),\\ u_x(1, t)= u(1, t)^q\quad &\text{on }[0,T),\\ u_x(0, t)= 0\quad &\text{on }[0, T),\\ u(x, 0)= u_0(x)\geq 0\quad &\text{in }[0,1]\end{cases}\tag{1} \] is studied, where \(p, q> 1\) and \(\lambda> 0\) are parameters. For this type of problems, existence and regularity of solutions have been proved. The following is known: 1) If \(p< 2q-1\) or \(p= 2q-1\) with \(\lambda< q\) and \(u_0> v\), where \(v\) is any maximal stationary solution; then \(u\) blows up in finite time. 2) If \(p> 2q-1\) or \(p= 2q-1\) with \(\lambda\geq q\) then every positive solution is global.
The numerical semidiscrete version of (1) proposed here comes from a first-order finite element approximation on the variable \(x\) with an uniform mesh keeping \(t\) continuous. We obtain a system: \[ \begin{cases} u_1'= {2\over h^2} (u_2- u_1)- \lambda u^p_1,\\ u_k'= {1\over h^2} (u_{k+ 1}- 2u_k+ u_{k-1})-\lambda u^p_,\quad &2\leq k\leq N-1,\\ u_N'= {2\over h^2} (u_{N- 1}- u_N)-\lambda u^q_N+{2\over h} u^q_N,\\ u_i(0)= u_0(x_i),\quad &1\leq i\leq N,\end{cases}\tag{2} \] where \(x_i\) is an uniform partition of the interval \([0,1]\).
The following convergence theorem holds: Theorem 1.2. Let \(u\in C^{2,1}([0, 1]\times [0, T_1])\) be the solution of (1) and \(u_h\) its semidiscrete approximation. Then there exists a constant \(C\) depending on \(T_1\) and \(u\) such that, for \(h\) small enough: \[ \|u- u_h\|_{L^\infty([0, 1]\times [0, T_1])}\leq Ch^{{3\over 2}}. \] For a semidiscrete system (2) the following theorem describing when the blow-up phenomenon occurs is proved: Theorem 1.3. Let \(U= (u_1,\dots, u_N)\) be a positive solution of (2). Then: 1. if \(p\leq q\) and the initial datum is large enough, \(U\) has finite blow-up time; 2. if \(p> q\), \(U\) is global.
The different behaviour of the solutions in the continuous and semidiscrete case is explained. The reason for this is that a spurious attractive steady solution appears which goes to infinity as \(h\) goes to zero.