.NET Core 1.0 was released on June 27 2016 and .NET Core 1.1 was released on November 16 2016. As an LTS release, .NET Core 1.0 is supported for 3 years and .NET Core 1.1 fits into the same support timeframe as .NET Core 1.0.
So in short : .NET Core 1.0 and 1.1 will reach end of life and go out of support on June 27, 2019.
After this date of June 27, 2019, .NET Core updates will no longer include updated packages or Docker container images for .NET Core 1.0 and 1.1. But wait, no panic and this is for the best, upgrade from .NET Core 1.x to .NET Core 2.1 or 2.2.
.NET Core 2.1 is a long-term support (LTS) release so makes it your minimum standard for your development, even if I encourage you to use .NET Core 2.2. Why .NET Core 2.1 at least and not .NET Core 2.0 ? Because .NET Core 2.0 has already reached end-of-life, as of October 1, 2018
Upgrade to .NET Core 2.x
The migration is quick and easy but be sure to bulletproof your application with heavy testing, I already found several minor issues while migrating from 1.1 to 2.1.