# Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆

ESAIM: Proceedings (2011)

- Volume: 31, page 73-100
- ISSN: 1270-900X

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topRubio, Gerardo. Emilia Caballero, Ma., et al, eds. "Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆." ESAIM: Proceedings 31 (2011): 73-100. <http://eudml.org/doc/251228>.

@article{Rubio2011,

abstract = {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.},

author = {Rubio, Gerardo},

editor = {Emilia Caballero, Ma., Chaumont, Loïc, Hernández-Hernández, Daniel, Rivero, Víctor},

journal = {ESAIM: Proceedings},

keywords = {Cauchy problem; semilinear parabolic partial differential equations; stochastic control problems; approximation with linear parabolic equations},

language = {eng},

month = {3},

pages = {73-100},

publisher = {EDP Sciences},

title = {Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆},

url = {http://eudml.org/doc/251228},

volume = {31},

year = {2011},

}

TY - JOUR

AU - Rubio, Gerardo

AU - Emilia Caballero, Ma.

AU - Chaumont, Loïc

AU - Hernández-Hernández, Daniel

AU - Rivero, Víctor

TI - Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆

JO - ESAIM: Proceedings

DA - 2011/3//

PB - EDP Sciences

VL - 31

SP - 73

EP - 100

AB - 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.

LA - eng

KW - Cauchy problem; semilinear parabolic partial differential equations; stochastic control problems; approximation with linear parabolic equations

UR - http://eudml.org/doc/251228

ER -

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