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Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms. (English) Zbl 0757.35034
This paper deals with Hamilton-Jacobi equations of the form $$(SP) \quad u+\langle Ax,Du\rangle+F(x,u,Du)=0, \qquad (CP) \quad u\sb t+\langle Ax,Du\rangle+F(t,x,u,Du)=0,$$ where $A$ is an $m$-accretive operator on a Banach space $X$, $A: D(A)\to 2\sp X$ and $F:\overline{D(A)}\times R\times X\sp x\to R$ (respectively $F:\{0,T\}\times\overline{D(A)}\times R\times X\sp x\to R$) are continuous mappings. The author presents a new definition of viscosity solution for the Hamilton-Jacobi equations $(SP)$ and $(CP)$ and proves existence and uniqueness results under usual assumptions on the bounded part of the Hamiltonian.

##### MSC:
 35K55 Nonlinear parabolic equations 49L25 Viscosity solutions (infinite-dimensional problems) 70H20 Hamilton-Jacobi equations (mechanics of particles and systems)
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##### References:
 [1] Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. (1976) · Zbl 0328.47035 [2] Barbu, V.; Da Prato, G.: Hamilton-Jacobi equations in Hilbert spaces. Research notes in mathematics 86 (1983) · Zbl 0508.34001 [3] Crandall, M. G.; Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. amer. Math. soc. 277, 1-42 (1983) · Zbl 0599.35024 [4] Crandall, M. G.; Lions, P. L.: Hamilton-Jacobi equations in infinite dimensions. III. existence of viscosity solutions. J. funct. Anal. 68, 214-247 (1986) · Zbl 0739.49015 [5] Crandall, M. G.; Lions, P. L.: Viscosity solutions of Hamilton-Jacobi equation in infinite dimensions. IV. Hamiltonians with unbounded linear terms. J. funct. Anal. 87 (1989) · Zbl 0739.49016 [6] Crandall, M. G.; Evans, L. C.; Lions, P. L.: Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. amer. Math. soc. 282, 487-502 (1984) · Zbl 0543.35011 [7] Evans, L. C.: Some MIN-MAX methods for the Hamilton-Jacobi equations. Indiana univ. Math. J. 33, 31-50 (1984) · Zbl 0543.35012 [8] Evans, L. C.; Souganidis, P.: Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaac equations. Indiana univ. Math. J. 33, 721-748 (1984) [9] Ishii, H.: Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations. Indiana univ. Math. J. 33, 773-797 (1984)