A Remark on Parabolic Equations

Alain Haraux

Abstract: If $L=L^{*}$ is a seif-adjoint linear operator generating a strongly continuous semi-group on a real Hilbert space $H$ and $\alpha\in L^{\infty}(\R^{+})$, any mild solution $u$ of $u'=Lu+\alpha(t)\,u$ satisfies $(u(0),u(t))\ge0$ for all $t\ge0$. On the other hand for any $\lambda>0$ such that $(\pi/L)^{2}<\lambda<4(\pi/L)^{2}$, there are solutions $u$ of the one-dimensional semilinear heat equation $u_{t}-u_{xx}+u^{3}-\lambda u=0$ in $\R^{+}\times(0,L)$, $u(t,0)=u(t,L)=0$ on $\R^{+}$ such that $\int_{\Omega}u(0,x)\,u(t,x)\,dx<0$ for some $t>0$.

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