In this paper, the adomian decomposition method is applied to nonlinear heat equation with exponential nonlinearity. This method is tested for some examples. The results obtained show that the method is efficient and accurate. This study showed also, the speed of the convergent of Adomian decomposition method.

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Mathematics Subject Classification (1991): 31C45, 31D05, 35G30, 35K60 0. Introduction Let us consider a locally compact space X and a nonlinear sheaf H on X. We define a (nonlinear) harmonic Bauer space (X, H) in a way which generalizes most of all previous settings [5], [12], [16], [20], [21]. We then define hyperharmonic and hypoharmonic functions related to such a nonlinear harmonic space. Following Serrin [23] and Lehtola [21], we introduce the Harnack principle or the Harnack inequality (H) as follows: For every open domain U in X and every compact subset K of U there exist two positive constants c 1 and c 2 such that sup K h ≤ c 1 infK h + c 2 for every h ∈ H + (U ). Using semi-linear perturbations, we find several examples of Bauer spaces where the constants c 1 and c 2 in (H ) are necessarily both strictly positive and hence the classical Harnack inequality (sup K h ≤ c inf K h) does not hold. We even find several examples where we can choose c 1 = 0. Throughout this paper we discuss the validity of the convergence property of Brelot and we prove that it does not imply ellipticity in contrast to the linear case. Let U be an open subset of R d , V an open subset of R d × R, d ≥ 1, and α a positive real number. We consider: H α (U ) := {u ∈ C(U ) : ∆u = sgn(u)|u| α in the distributional sense} H α (V ) := u ∈ C(V ) : ∆u − ∂u ∂t = sgn(u)|u| α in the distributional sense

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