On the asymptotic properties for simple semilinear heat equations (Q799866)

The behavior of solutions of the Cauchy problem (as \(t\to\infty ) (1)\quad u_ t(t,x)=\Delta u(t,x)+f(u(t,x)),\quad t>0,\quad x\in R^ N,\quad N\geq 3,\quad (2)\quad u(0,x)=a(x),\quad x\in R^ N,\) is investigated, where the function f is defined as follows: \(f(\lambda)=p\lambda -pq\) when \(\lambda \geq q\), \(f(\lambda)=0\) when 0\(\leq\lambda \leq q\); p,q are positive constants. It is proved that this asymptotic behavior depends on the initial value a(x). The main result: Let \(\ell\) be a constant with 0\(\leq\ell \leq q\) and suppose that a(x) is a non-negative bounded continuous function. Let \(r=| x|\). (i) If \(\ell\leq a(x)\leq u_{\ell}(r)\) and \(a(x)\not\equiv u_{\ell}(r)\), then the solution u(t,x,a,f) of (1) with (2) converges to \(\ell\) uniformly in x as \(t\to\infty \). (ii) If \(a(x)\geq u_{\ell}(r)\), then the solution u(t,x,a,f) of (1), (2) gives up to infinity as \(t\to\infty \). \(u_{\ell}(r)\) is the solution of the radially symmetric problem \(u''(r)+(N-1/r)u'+f(u)=0,\quad r>0,\quad u(0)>0,\quad u'(0)=0\) with \(u(0)=q\) and \(\lim_{r\to\infty }u_{\ell}(r)=\ell.\)