Розглянуто нелінійне параболічне рівняння, збурене білим шумом. Доведено існування та єдиність узагальненого розв'язку мішаної задачі для цього рівняння.

Let p ( · ) $p(\cdot )$ be a measurable function defined on R d ${\mathbb {R}}^d$ and − : = inf x ∈ $p_-:=\inf _{x\in {\mathbb {R}}^d}p(x)$ . In this paper, we generalize the Hardy–Littlewood maximal operator. definition, instead of cubes or balls, take supremum over all rectangles side lengths which are in cone-like set by given ψ. Moreover, integral means, consider L q $L_{q(\cdot )}$ -means....

In this work, approximation properties of de la Vall?e-Poussin means are investigated in weighted Lebesgue spaces with variable exponent where weight function belongs to Muckenhoupt class. For purpose direct, inverse and simultaneous theorems theory proved constructive characterizations functions obtained exponent.

In this work, we investigate the following Kirchhoff-type equation with variable exponent nonlinearities u_{tt}-M(‖∇u‖²)△u+|u_{t}|^{p(x)-2}u_{t}=|u|^{q(x)-2}u. We proved the blow up of solutions in finite time by using modified energy functional method.