Abstract

In this work, we prove existence local entropy solution of a convection-diffusion type integro-differential equation \displaystyle \partial_{t}\bigg(k* (j(v)-j(v_{0}))\bigg) - \nabla\cdot\bigg( a(x,\nabla \varphi (v))+ F(\varphi (v))\bigg) = f in $Q_{T}:= (0,T) \times \Omega$ with Dirichlet boundary condition $v(0, \cdot{})= v_{0}$ in $\Omega$ and $L^{1}$-data $f \in L^{1}((0,T)\times \Omega), \ j(v_{0})\in L^{1}(\Omega)$. To that end, regularising the data by $L^{\infty}$-functions, using the existence result of entropy solution for these more approximate data and a comparison and diagonal principle of the regularised entropy solution, we prove the existence of an local entropy solution.

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