Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with \(t\). Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time \(T^\ast\). Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span \(T^\ast\) is derived by the means of integro-differential inequality techniques.

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equations . At low initial energy level ( ), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.

The sliding window method is a commonly used data processing in time series forecasting tasks, and determining the appropriate window size is a crucial step in constructing predictive models. However, the current setting of window size parameters is often based on empirical knowledge, making the scientific determination of the optimal sliding window size highly significant. This paper proposes an algorithm for optimizing window size based on sample entropy, which is applicable not only to the original undecomposed sequences but also effectively to the decomposed sequences. The proposed algorithm has been validated using the open-source Elia grid data across multiple model architectures, including recurrent (GRU/LSTM) and attention-based (Transformer) networks. Experimental results demonstrate that the algorithm effectively determines an optimal window size of 106. The optimized window consistently leads to superior prediction performance, with the CEEMD-GRU model achieving a MAPE of 0.256, RMSE of 22.529, and MAE of 18.186—representing reductions of over 5% compared to the undecomposed benchmark. Furthermore, the enhancement is more significant for decomposed sequences, and the algorithm’s efficacy is validated across different neural network architectures (e.g., LSTM, GRU, Transformer), confirming its practical utility and generalizability.

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source u(t) - Delta u(m) = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.