Alvino, Angelo; Volpicelli, Roberta; Volzone, Bruno Sharp estimates for solutions of parabolic equations with a lower order term. (English) Zbl 1173.35029 J. Appl. Funct. Anal. 3, No. 1, 61-88 (2008). The interesting paper under review deals with comparison results for solutions to Cauchy-Dirichlet problems for parabolic equations by means of Schwarz symmetrization. Precisely, the authors consider weak solutions \(u(x,t)\) and \(v(x,t)\) respectively of the problems \[ \begin{cases} u_t-\sum_{i,j=1}^N \big(a_{ij}(x,t)u_{x_i}\big)_{x_j}+c(x)u=f & \text{in}\;\Omega\times(0,T),\\ u=0 & \text{on}\;\partial\Omega\times(0,T),\\ u(x,0)=u_0(x), & x\in\Omega \end{cases} \]and \[ \begin{cases} v_t-\Delta v+c_{\#} v=f^{\#} & \text{in}\;\Omega^{\#}\times(0,T),\\ v=0 & \text{on}\;\partial\Omega^{\#}\times(0,T),\\ v(x,0)=u_0^{\#}(x), & x\in\Omega^{\#}. \end{cases} \]Here \(\Omega\subset\mathbb R^N\) is a bounded and open set, \(\Omega^{\#}\) is the \(N\)-dimensional ball centered at the origin and having the same measure as \(\Omega,\) while \(c_{\#}\) and \(f^{\#}\) are respectively the increasing spherical rearrangement of \(c\) and the decreasing spherical rearrangement of \(f.\) Assuming uniform parabolicity of the divergence-form operator above with \(a_{ij}\in L^\infty(\Omega\times(0,T)),\) \[ c\in L^r(\Omega)\;\text{ with }\;r>N/2\;\text{ if }\;N\geq2,\quad r\geq1\;\text{ if }\;N=1, \]\(f\in L^2(\Omega\times(0,T))\) and \(u_0\in L^2(\Omega),\) the authors prove that for all \(t\in[0,T],\) the decreasing rearrangements \(u^*\) and \(v^*\) of \(u\) and \(v,\) respectively, satisfy the inequality \[ \int_0^s u^*(\sigma,t)\,d\sigma \leq \int_0^s v^*(\sigma,t)\,d\sigma\quad \forall s\in \big[0,|\Omega|\big]. \] Reviewer: Lubomira Softova (Aversa) Cited in 2 Documents MSC: 35B45 A priori estimates in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs Keywords:rearrangements; comparison results; Schwarz symmetrization; Cauchy-Dirichlet problems; divergence-form operator PDFBibTeX XMLCite \textit{A. Alvino} et al., J. Appl. Funct. Anal. 3, No. 1, 61--88 (2008; Zbl 1173.35029)