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On the strong maximum principle for second order nonlinear parabolic integro-differential equations. (English) Zbl 1264.35262
In this paper the author investigates the strong maximum principle for viscosity solutions of second-order nonlinear parabolic integro-differential equations of the form $u_{t}+F\left( x,t,Du,D^{2}u,\mathcal{I}\left[ x,t,u\right] \right) =0\text{ in }\Omega \times \left( 0,T\right)$ where $$\Omega \subset \mathbb{R}^{N}$$ is an open bounded set, $$T>0,$$ and $$u$$ is a real-valued function defined on $$\mathbb{R}^{N}\times \left[ 0,T\right]$$ and is assumed that the function $$u=u\left( x,t\right)$$ is a priori defined outside the domain $$\Omega$$. The nonlinearity $$F$$ is a real-valued, continuous function in $$\Omega \times \left[ 0,T\right] \times \mathbb{R}^{N}\times \mathbb{S}^{N}\times \mathbb{R} ,\mathbb{S}^{N}$$ being the set of real symmetric $$N\times N$$ matrices. $$F$$ is considered to be degenerate elliptic, i.e. $F\left( x,t,p,X,l_{1}\right) \leq F\left( x,t,p,Y,l_{2}\right) ,X\geq Y,l_{1}\geq l_{2},$ for all $$\left( x,t\right) \in \overline{\Omega }\times \left[ 0,T\right] ,p\in \mathbb{R}^{N}\backslash \left\{ 0\right\} ,X,Y\in \mathbb{S} ^{N},l_{1},l_{2}\in \mathbb{R.}$$ $$\mathcal{I}\left[ x,t,u\right]$$ is an integro-differential operator of the type $\mathcal{I}\left[ x,t,u\right] =\int_{\mathbb{R}^{N}}\left( u\left( x+z,t\right) -u\left( x,t\right) -Du\left( x,t\right) \cdot z1_{B}\left( z\right) \right) \mu _{x}\left( dz\right) ,$ where $$1_{B}\left( z\right)$$ denotes the indicator function of the unit ball $$B$$ and $$\left\{ \mu _{x}\right\} _{x\in \mathbb{\Omega }}$$ is a family of Levy measures.
In particular, $\mathcal{I}\left[ x,t,u\right] =\int_{\mathbb{R}^{N}}\left( u\left( x+j\left( x,z\right) ,t\right) -u\left( x,t\right) -Du\left( x,t\right) \cdot j\left( x,z\right) 1_{B}\left( z\right) \right) \mu \left( dz\right) ,$ is a Levy-Ito operator, where $$j\left( x,z\right)$$ is the size of the jumps at $$x$$ satisfying $j\left( x,z\right) \leq C_{0}\left| z\right| ,\forall x\in \Omega ,\forall z\in \mathbb{R}^{N},$ with $$C_{0}$$ a positive constant.
If $$Q_{T}=\Omega \times \left( 0,T\right]$$, then for every point $$P_{0}=\left( x_{0},t_{0}\right) \in Q_{T},S\left( P_{0}\right)$$ is the set of all points $$Q\in Q_{T}$$ which can be connected to $$P_{0}$$ by a simple continuous curve in $$Q_{T}$$ and $$C\left( P_{0}\right)$$ denotes the connected component of $$\Omega \times \left\{ t_{0}\right\}$$ which contains $$P_{0}$$. In Section 2 of the paper are studied, separately, the propagation of maxima in $$C\left( P_{0}\right)$$ and in the region $$\Omega \times \left( 0,t_{0}\right)$$. In Section 3 similar results are given in the case of Levy-Ito operators. Examples are provided in Section 4. In Section 5 the author proves a strong comparison result for the Dirichlet problem, based on the strong maximum principle for the linearized equation.

##### MSC:
 35R09 Integro-partial differential equations 35K55 Nonlinear parabolic equations 35B50 Maximum principles in context of PDEs 35D40 Viscosity solutions to PDEs 35B51 Comparison principles in context of PDEs
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