Geodesic
Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions
Gerardo Rubio1
1 Departamento de Matemáticas, Facultad de Ciencias, UNAM, MÉXICO
ESAIM. Proceedings, Tome 31 (2011), pp. 73-100.

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We consider the Cauchy problem in ℝd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coefficients for the semilinear equation. Our approach is using stochastic differential equations and parabolic differential equations in bounded domains.Finally, we apply the results to a stochastic optimal consumption problem.
DOI : 10.1051/proc/2011005

Gerardo Rubio 1

1 Departamento de Matemáticas, Facultad de Ciencias, UNAM, MÉXICO 
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     author = {Gerardo Rubio},
     title = {Existence and uniqueness to the {Cauchy} problem for linear and semilinear parabolic equations with local conditions},
     journal = {ESAIM. Proceedings},
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     year = {2011},
     doi = {10.1051/proc/2011005},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/2011005/}
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Gerardo Rubio. Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions. ESAIM. Proceedings, Tome 31 (2011), pp. 73-100. doi : 10.1051/proc/2011005. http://geodesic.mathdoc.fr/articles/10.1051/proc/2011005/

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