### A decomposition method for a semilinear boundary value problem with a quadratic nonlinearity.

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The operator ${L}_{0}:{D}_{{L}_{0}}\subset H\to H$, ${L}_{0}u=\frac{1}{r}\frac{d}{dr}\left\{r\frac{d}{dr}\left[\frac{1}{r}\frac{d}{dr}\left(r\frac{du}{dr}\right)\right]\right\}$, ${D}_{{L}_{0}}=\{u\in {C}^{4}\left([0,R]\right),{u}^{\text{'}}\left(0\right)={u}^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)=0,u\left(R\right)={u}^{\text{'}}\left(R\right)=0\}$, $H={L}_{2,r}(0,R)$ is shown to be essentially self-adjoint, positive definite with a compact resolvent. The conditions on ${L}_{0}$ (in fact, on a general symmetric operator) are given so as to justify the application of the Fourier method for solving the problems of the types ${L}_{0}u=g$ and ${u}_{tt}+{L}_{0}u=g$, respectively.

We construct Almansi decompositions for a class of differential operators, which include powers of the classical Laplace operator, heat operator, and wave operator. The decomposition is given in a constructive way.

We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.

We give necessary and sufficient conditions for the formal power series solutions to the initial value problem for the Burgers equation ${\partial}_{t}u-{\partial}_{x}\xb2u={\partial}_{x}\left(u\xb2\right)$ to be convergent or Borel summable.

Let $L(z,{\partial}_{z})=({\partial}_{{z}_{0}}{)}^{k}-A(z,{\partial}_{z})$ be a linear partial differential operator with holomorphic coefficients, where$$A(z,{\partial}_{z})=\sum _{j=0}^{k-1}{A}_{j}(z,{\partial}_{{z}^{\text{'}}}\left)\right({\partial}_{{z}_{0}}{)}^{j},\phantom{\rule{3.33333pt}{0ex}}\mathrm{ord}.A(z,{\partial}_{z})=m\>k$$and$$z=({z}_{0},{z}^{\text{'}})\in {C}^{n+1}.$$We consider Cauchy problem with holomorphic data$$L(z,{\partial}_{z})u\left(z\right)=f\left(z\right),\phantom{\rule{3.33333pt}{0ex}}\left({\partial}_{{z}_{0}}{)}^{i}u\right(0,{z}^{\text{'}})={\widehat{u}}_{i}({z}^{\text{'}}\left)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\right(0\le i\le k-1).$$We can easily get a formal solution $\widehat{u}\left(z\right)={\sum}_{n=0}^{\infty}{\widehat{u}}_{n}({z}^{\text{'}}\left)\right({z}_{0}{)}^{n}/n!$, bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L(z,{\partial}_{z})$, there is a function ${u}_{S}\left(z\right)$ holomorphic except on $\{{z}_{0}=0\}$ such that $L(z,{\partial}_{z}){u}_{S}\left(z\right)=f\left(z\right)$ and ${u}_{S}\left(z\right)\sim \widehat{u}\left(z\right)$ as ${z}_{0}\to 0$ in $S$.