The global feature of unbounded solutions to a nonlinear parabolic equation (Q1194503)

The purpose of this paper is to study the blow-up set and asymptotic behaviour of the solution of the Cauchy problem \[ u_ t=u^ \alpha(u_{xx}+u),\qquad x\in R^ 1,\quad t\geq 0, \tag{1} \] \(u(x,0)=u_ 0(x)\geq 0\), \(x\in R^ 1\), where \(\alpha>0\), \(u_ 0(x)\neq 0\) is a continuous function with compact support. The main results are the following: Theorem 3. Let \(0<\alpha<2\) and let \(u_ 0(x)\geq 0\) be a function which satisfies: \(\text{supp}u_ 0(x)\supset(a,b)\). If \(b-a>2\pi/\alpha\), then the solution \(u(x,t)\) of (1) will blow-up in finite time. Theorem 4. Let \(u(x,t)\) be a solution of (1) which blows-up in finite time \(T<\infty\). If \(u_ 0(x)\in C^ 1(R^ 1)\) satisfies: \(u_ 0(- x)=u_ 0(x)\), \(x\in R^ 1\), \(u_ 0(x)\) decreasing for \(x>0\) and \(\sup u_ 0(x)=u(0)\), then the following holds: \((T-t)^{1/\alpha}u(x,t)\to w(x)\) as \(t\to T\), uniformly in \(R^ 1\), where \(w\) is a similarity solution of the equation (1).