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**Existence of positive solutions due to non-local interactions in a class of nonlinear boundary value problems.**
*(English)*
Zbl 1149.35366

From the text: We study the steady states of a class of non-local differential equations of the form

\[ u_t(x,t)= \sum_{i=1}^n (a_i(x)u_{x_i}(x,t))_{x_i}+ b(x)u(x,t)+\varepsilon \tilde{u}(t)+ n(x,\varepsilon,u(x,t),\tilde{u}(t)), \quad x\in U,\;t>0; \]

\[ u(x,t)=0 \quad\text{for \(x\) on }\partial U,\;t\geq 0; \qquad u(x,0)= u_0(x), \tag{1} \]

where \(U\) is a bounded connected subset of \(\mathbb R^n\), \(n=1,2,3\) with a suitably smooth boundary, \(\partial U\), \(\varepsilon\in\mathbb R^+\), \(\tilde{u}(t):= \int_U u(x,t)\,dx\), and \(n:U\times\mathbb R\times\mathbb R\times\mathbb R\to\mathbb R\) with \(n(x,\varepsilon,\zeta,\tilde{\zeta})= o(\zeta)\) for \(\zeta\) near zero, uniformly in bounded \(\varepsilon\) intervals. Moreover, we assume that \(n(x,0,u,\widetilde{u})= f(x,u)\) for some suitable function \(f\).

Equation (1) is referred to as a non-local differential equation, as it contains the functional \(\widetilde{u}\), the value of which depends upon the value of \(u(x)\) the domain \(U\). The parameter \(\varepsilon\) can be considered as a measure of the strength of the non-local effects in the equation: for \(\varepsilon=0\), the final condition on \(n\) given above, ensures that (1) reduces to a local differential equation.

We consider a class of non-local boundary value problems of the type used to model a variety of physical and biological processes, from Ohmic heating to population dynamics. Of particular relevance therefore is the existence of positive solutions. We are interested in the existence of such solutions that arise as a direct consequence of the non-local interactions in the problem. Conditions are therefore imposed that preclude the existence of a positive solution for the related local problem. Under these conditions, we prove that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise.

\[ u_t(x,t)= \sum_{i=1}^n (a_i(x)u_{x_i}(x,t))_{x_i}+ b(x)u(x,t)+\varepsilon \tilde{u}(t)+ n(x,\varepsilon,u(x,t),\tilde{u}(t)), \quad x\in U,\;t>0; \]

\[ u(x,t)=0 \quad\text{for \(x\) on }\partial U,\;t\geq 0; \qquad u(x,0)= u_0(x), \tag{1} \]

where \(U\) is a bounded connected subset of \(\mathbb R^n\), \(n=1,2,3\) with a suitably smooth boundary, \(\partial U\), \(\varepsilon\in\mathbb R^+\), \(\tilde{u}(t):= \int_U u(x,t)\,dx\), and \(n:U\times\mathbb R\times\mathbb R\times\mathbb R\to\mathbb R\) with \(n(x,\varepsilon,\zeta,\tilde{\zeta})= o(\zeta)\) for \(\zeta\) near zero, uniformly in bounded \(\varepsilon\) intervals. Moreover, we assume that \(n(x,0,u,\widetilde{u})= f(x,u)\) for some suitable function \(f\).

Equation (1) is referred to as a non-local differential equation, as it contains the functional \(\widetilde{u}\), the value of which depends upon the value of \(u(x)\) the domain \(U\). The parameter \(\varepsilon\) can be considered as a measure of the strength of the non-local effects in the equation: for \(\varepsilon=0\), the final condition on \(n\) given above, ensures that (1) reduces to a local differential equation.

We consider a class of non-local boundary value problems of the type used to model a variety of physical and biological processes, from Ohmic heating to population dynamics. Of particular relevance therefore is the existence of positive solutions. We are interested in the existence of such solutions that arise as a direct consequence of the non-local interactions in the problem. Conditions are therefore imposed that preclude the existence of a positive solution for the related local problem. Under these conditions, we prove that there exists a unique positive solution to the boundary value problem for all sufficiently strong non-local interactions and no positive solutions exists otherwise.