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Sharp estimates for solutions of parabolic equations with a lower order term. (English) Zbl 1173.35029

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
\[ \int_0^s u^*(\sigma,t)\,d\sigma \leq \int_0^s v^*(\sigma,t)\,d\sigma\quad \forall s\in \big[0,|\Omega|\big]. \]

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
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