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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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Full Text: Euclid