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Uniqueness and asymptotics of singularly perturbed equations involving implicit boundary conditions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

A class of one-dimensional convection–diffusion equations with a singularly perturbed parameter in a bounded domain is presented, where the boundary condition is nonlocal type with an implicit form related to the unknown solutions. In general, the validity of the maximum principle of this type equation is unassurable. Employing a singular perturbations method as a natural tool, we establish the uniqueness and maximum principle as the singularly perturbed parameter is sufficiently small. Such an argument is different from the standard fixed point approaches. Moreover, as this parameter tends to zero, the boundary and interior asymptotics of solutions is obtained.

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Notes

  1. Note that the equation of \(v_{\varepsilon ,\lambda }\) is equivalent to the equation \(-\varepsilon ^2(D_{\varepsilon }(x)v_{\varepsilon ,\lambda }'(x))'+D_{\varepsilon }(x)b(x)f(v_{\varepsilon ,\lambda })=0\), \(x\in (0,\ell )\), with the corresponding energy functional \(E_{\varepsilon }[v_{\varepsilon ,\lambda }]=\int _0^{\ell }D_{\varepsilon }\left( \frac{\varepsilon ^2}{2}v_{\varepsilon ,\lambda }'^2+bF(v_{\varepsilon ,\lambda })\right) \,\mathrm {d}x\), where \(D_{\varepsilon }(x):=\exp \left( -\int _0^x\frac{a(y)}{\varepsilon }\,\mathrm {d}y\right) \) is positive on \([0,\ell ]\), and \(F(t)=\int _0^tf(s)\,{\mathsf {d}}s\ge 0\) is convex (by (1.4)). Since \(b>0\), for each fixed \(\varepsilon >0\) and \(\lambda \in {\mathbb {R}}\), by the direct method in the calculus of variations one obtains that \(E_{\varepsilon }[v_{\varepsilon ,\lambda }]\) is a convex functional and has a minimizer over the space \({\mathcal {H}}=\{v_{\varepsilon ,\lambda }\in \text {H}^1((0,\ell )):\,v_{\varepsilon ,\lambda }(0)=0,\,\,v_{\varepsilon ,\lambda }(\ell )=\lambda \}\). As a consequence, the regularity theory for elliptic equations implies that this minimizer is a classical solution of (1.6).

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Acknowledgements

The author would like to thank the anonymous reviewers for the constructive comments that contributed to enhancing the overall quality of the original manuscript. This work was partially supported by the MOST grants of Taiwan with numbers 108-2115-M-007-006-MY2 and 110-2115-M-007 -003 -MY2.

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  1. Institute for Computational and Modeling Science, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Chiun-Chang Lee

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Lee, CC. Uniqueness and asymptotics of singularly perturbed equations involving implicit boundary conditions. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 51 (2023). https://doi.org/10.1007/s13398-022-01383-6

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