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Logarithmic upper bounds for weak solutions to a class of parabolic equations
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 149 / Issue 6 / December 2019
- Published online by Cambridge University Press:
- 16 January 2019, pp. 1481-1491
- Print publication:
- December 2019
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- Article
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It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation
$\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in 
$\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate
provided that
$$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$
$q \gt 1+{N}/{2}$, where Ω is a bounded domain in 
${\open R}^N$ with Lipschitz boundary, T > 0, 
$\partial _p\Omega _T$ is the parabolic boundary of 
$\Omega _T$, 
$\omega \in L^1(\Omega _T)$ with 
$\omega \ges 0$, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if 
$q=1+{N}/{2}$. In this paper, we show that the linear growth of the upper bound in 
$\Vert f \Vert_{q,\Omega _T}$ can be improved. To be precise, we establish
$$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$
