Prepare the Crust:**
- In a medium bowl, mix the graham cracker crumbs, melted butter, and sugar until the crumbs are evenly moistened.
- Press the mixture firmly into the bottom of a 9-inch springform pan to form an even crust. Place the crust in the refrigerator to set while you prepare the filling.
**Prepare the Blackberry Sauce:**
- In a medium saucepan, combine the blackberries, sugar, and lemon juice. Cook over medium heat until the blackberries break down and release their juices, about 5-7 minutes.
- Stir in the cornstarch mixture and continue to cook until the sauce thickens, about 2-3 minutes.
- Remove the sauce from the heat and let it cool completely. Once cooled, strain the sauce through a fine-mesh sieve to remove the seeds, if desired.
**Prepare the Cheesecake Filling:**
- In a large mixing bowl, beat the softened cream cheese, powdered sugar, and vanilla extract together until smooth and creamy.
- In a separate bowl, whip the heavy cream until stiff peaks form.
- Gently fold the whipped cream into the cream cheese mixture until fully combined and smooth.
**Assemble the Cheesecake:**
- Pour half of the cheesecake filling over the prepared crust and spread it evenly.
- Drizzle a few tablespoons of the blackberry sauce over the filling, then swirl it gently with a knife or skewer.
- Add the remaining cheesecake filling on top and smooth the surface.
- Drizzle more blackberry sauce on top and create another swirl pattern for a decorative finish.
**Chill the Cheesecake:**
- Cover the cheesecake with plastic wrap and refrigerate for at least 6 hours or overnight to allow it to set completely.
**Serve:**
- Once set, carefully remove the cheesecake from the springform pan and transfer it to a serving plate.
- Garnish with fresh blackberries and additional blackberry sauce if desired.
- Slice and serve chilled.